Geometry

Problem 2 Finding Circumference

Which formula should What is the circumference of the circle in terms of P? What is the circumference of
y ou use? the circle to the nearest tenth?
You want to find
circumference. Since A }M
you know the diameter
in part (A), it would C = pd Use the formula for circumference of a circle. 15 in.
be easier to use the This is the exact answer. M
circumference formula = p(15) Use a calculator.
that involves diameter.
≈ 47.1238898

The circumference of }M is 15p in., or about 47.1 in.

B }T Use the formula for circumference of a circle. 4 cm
C = 2pr hsm11gmse_0108_t01191.ai

= 2p(4) This is the exact answer. T

= 8p Simplify.

≈ 25.13274123 Use a calculator.

The circumference of }T is 8p cm, or about 25.1 cm.

Got It? 2. a. What is the circumference of a circle with radius 24 m in terms ofhps?m11gmse_0108_t01

b. What is the circumference of a circle with diameter 24 m to the
nearest tenth?

Problem 3 Finding Perimeter in the Coordinate Plane

What do you need? Coordinate Geometry  What is the perimeter of △EFG? y
To find the perimeter of a Step 1 Find the length of each side.
figure, you need its side 6 E(3, 6)
lengths. Use what you
know about length on a EF = 0 6 - ( -2) 0 = 8 Use the Ruler Postulate. 4
number line and in the FG = 0 3 - (-3) 0 = 6
coordinate plane.
EG = 2(3 - ( - 3))2 + (6 - ( - 2))2 Use the Distance
Formula. Ϫ3 O x
G(؊3, ؊2)
= 262 + 82 Simplify within the parentheses. 24
= 236 + 64 Simplify.
F(3, ؊2)

= 2100 hsm11gmse_0108_t01192.ai
= 10
Step 2 Add the side lengths to find the perimeter.
EF + FG + EG = 8 + 6 + 10 = 24

The perimeter of △EFG is 24 units.

Got It? 3. Graph quadrilateral JKLM with vertices J( -3, -3), K(1, -3), L(1, 4),
and M( -3, 1). What is the perimeter of JKLM?

Lesson 1-8  Perimeter, Circumference, and Area 61

To find area, you should use the same unit for both dimensions.

Problem 4 Finding Area of a Rectangle

Banners  You want to make a rectangular banner similar to the
one at the right. The banner shown is 221 ft wide and 5 ft high. To
the nearest square yard, how much material do you need?

Step 1 Convert the dimensions of the banner to yards. Use the
1 yd
conversion factor 3 ft .
#Width:
How can you check 5 ft 1 yd = 5 yd 221 ft = 5 ft
your conversion? 2 3 ft 6 2
Yards are longer than #Height: 5 ft
feet, so the number 1 yd = 5 yd
you get in yards should 3 ft 3
be less than the given
number in feet. Since Step 2 Find the area of the banner.
c56h6eck2s12. , the conversion
#A = bh Use the formula for area of a rectangle.

= 5 53 Substitute 5 for b and 5 for h.
6 6 3

= 25
18

The area of the banner is 1258 , or 1178 square yards (yd2). You need
2 yd2 of material.

Got It? 4. You are designing a poster that will be 3 yd wide and 8 ft high. How much
paper do you need to make the poster? Give your answer in square feet.

What are you given? Problem 5 Finding Area of a Circle 16 m
In circle problems, make What is the area of ⊙K in terms of P?
it a habit to note whether Step 1 Find the radius of ⊙K . K
you are given the radius
or the diameter. In this r = 126, or 8 The radius is half the diameter. hsm11gmse_0108_t01193.ai
case, you are given the Step 2 Use the radius to find the area.
diameter.
A = pr 2 Use the formula for area of a circle.
= p(8)2 Substitute 8 for r.
= 64p Simplify.

The area of ⊙K is 64p m2.

Got It? 5. The diameter of a circle is 14 ft.
a. What is the area of the circle in terms of p?
b. What is the area of the circle using an approximation of p?
c. Reasoning  Which approximation of p did you use in part (b)? Why?

62 Chapter 1  Tools of Geometry

The following postulate is useful in finding areas of figures with irregular shapes.

Postulate 1-10  Area Addition Postulate
The area of a region is the sum of the areas of its nonoverlapping parts.

Problem 6 Finding Area of an Irregular Shape

Multiple Choice  What is the area of the figure at the right? 3 cm
All angles are right angles.

27 cm2 45 cm2 9 cm
36 cm2 54 cm2

Step 1 Separate the figure into rectangles.

A1 3 cm
3 cm A1 3 cm 3 cm

A2 3 cm A2 3 cm hsm11gmse_0108_t01194.ai
6 cm A3 3 cm 6 cm

9 cm A3 3 cm
9 cm
What is another way
to find the area? Step 2 Find A1, A2, and A3.
Extend the figure to form
a square. Then subtract #hsAmrAe1a11==gm3bhse3_=0910 8_UStus0bes1tthi1teu9tfeo5rf.moarui tlahhefsobmratsh1ee1aangrdemahoesifegah_tr.0ec1ta0n8gl_e.t01196.ai
the areas of basic shapes #A2 = 6 3 = 18
from the area of the #A3 = 9 3 = 27
square.
Step 3 Find the total area of the figure.
A1
Total Area = A1 + A2 + A3 Use the Area Addition Postulate.
A A2 = 9 + 18 + 27
= 54
A = Asquare - A1 - A2

hsm11gmse_0108_tT0h1e7a0re3a.aoif the figure is 54 cm2. The correct choice is D.

Got It? 6. a. Reasoning  What is another way to separate the 4 ft
figure in Problem 6?

b. What is the area of the figure at the right?

12 ft

hsm11gmse_0108_t01197.ai

Lesson 1-8  Perimeter, Circumference, and Area 63

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

1. What is the perimeter and area of a rectangle with PRACTICES
base 3 in. and height 7 in.?
4. Writing  Describe a real-world situation in which

you would need to find a perimeter. Then describe a

2. What is the circumference and area of each circle to situation in which you would need to find an area.

the nearest tenth? 5. Compare and Contrast  Your friend can’t remember
whether 2pr computes the circumference or the area
a. r = 9 in. b. d = 7.3 m of a circle. How would you help your friend? Explain.

3. What is the perimeter 2y x 6. Error Analysis  A classmate finds the area of a circle
and area of the figure with radius 30 in. to be 900 in.2. What error did your
at the right? Ϫ4 O 4 classmate make?
Ϫ2

Practice and Probhlesmm1-1Sgomlsvei_n0g10E8_xt0e1r1c9is9e.asi MATHEMATICAL

A Practice Find the perimeter of each figure. PRACTICES See Problem 1.

7. 8.
4 in. 9 cm

7 in.

9. Fencing  A garden that is 5 ft by 6 ft has a walkway 2 ft wide around it. What is the

amount of fencing needed to surround the walkway? hsm11gmse_0108_t01201.ai

Findhtshme c1ir1cgummsfeer_e0nc1e0o8f_t}0C12in0t0e.rami s of P. See Problem 2.

10. 11. 5 ft 12. 13. 1 m
C 4
C C
15 cm C
3.7 in.

Coordinate Geometry  Graph each figure in the coordinate plane. Find See Problem 3.
each perimeter.

14. hXs(0m, 21)1, Yg(m4,s-e1_)0, Z1(0-82_,t-011)h2 s0m2.1a1i gmse_0108_1t50.1 hA2s(0m-44.1a, 1i-g1m), Bs(e4_, 05)1,0C8(4_,t-012h)2s0m6.1a1i gmse_0108_t01207.ai

16. L(0, 1), M(3, 5), N(5, 5), P(5, 1) 17. S( -5, 3), T(7, -2), U(7, -6), V( -5, -6)

Find the area of each rectangle with the given base and height. See Problem 4.
21. 40 cm, 2 m
18. 4 ft, 4 in. 19. 30 in., 4 yd 20. 2 ft 3 in., 6 in.

22. Roads  What is the area of a section of pavement that is 20 ft wide and 100 yd long?
Give your answer in square feet.

64 Chapter 1  Tools of Geometry

Find the area of each circle in terms of P. See Problem 5.
26.
23. 24. 3 25.
20 m 4 0.1 m
in. 6.3 ft

Find the area of each circle using an approximation of P. If necessary, round to the
nearest tenth.

27. rh=sm7 1ft1 gmse_0108_2t80.1 dh2s1=m08.1.a31img mse_0108_2t90.1 dh2s1=m32.14a1icgmm se_0108_3t0.1 rh2=s1m41.21ai1ng. mse_0108_t01215

Find the area of the shaded region. All angles are right angles. See Problem 6.

31. 20 m 32. 4 in. 33. 4 ft

18 m 8 in. 4 in. 8 ft
12 in. 8 ft
5 m 10 m
5m

B Apply Home Maintenance  To determine how much of each item to buy, tell whether you

need to know area or perimeter. Explain your choice.

34. hwsamllp1a1pgermfosrea_b0e1d0ro8o_mt0 1216.ai hsm11gmse3_50. 1cr0o8w_nt0m1o2ld1i7n.gafiorhascmei1li1ngg mse_0108_t01219.ai

36. fencing for a backyard 37. paint for a basement floor

38. Think About a Plan  A light-year unit describes the distance that one photon of light
travels in one year. The Milky Way galaxy has a diameter of about 100,000 light-years.
The distance to Earth from the center of the Milky Way galaxy is about 30,000 light-
years. How many more light-years does a star on the outermost edge of the Milky Way
travel in one full revolution around the galaxy compared to Earth?

• What do you know about the shape of each orbital path?
• Are you looking for circumference or area?
• How do you compare the paths using algebraic expressions?

39. a. What is the area of a square with sides 12 in. long? 1 ft long?
b. How many square inches are in a square foot?

40. a. Count squares at the right to find the area of the polygon 1 in.

outlined in blue.
b. Use a formula to find the area of each square outlined in red.
c. Writing  How does the sum of your results in part (b)

compare to your result in part (a)? Which postulate does

this support?

41. The area of an 11-cm-wide rectangle is 176 cm2. What is its length?

42. A square and a rectangle have equal areas. The rectangle is 64 cm by 81 cm.

What is the perimeter of the square? hsm11gmse_0108_t01218.ai

Lesson 1-8  Perimeter, Circumference, and Area 65

43. A rectangle has perimeter 40 cm and base 12 cm. What is its area?

Find the area of each shaded figure.

44. compact disc 45. drafting triangle 46. picture frame
50 mm 4 cm
2 in. 2 cm
5 cm
6 in. 7 cm

120 mm 3 in.
10 in.

47. a. Reasoning  Can you use the formula for the perimeter of a rectangle to find the

perimeter of any square? Explain.

bh. saCmnayn1r1yeogcutmaunssgeele_t?h0eE1xf0op8rlma_itnu0.la1f2o2r2th.aeiphesrimm1et1egr omf saes_qu0a1r0e8to_tfi0n1d2t2he3.paeirimhsemter1o1fgmse_0108_t01224.ai

c. Use the formula for the perimeter of a square to write a formula for the area of a

square in terms of its perimeter.

48. Estimation  On an art trip to England, a student sketches the 14 m 10 m
floor plan of the main body of Salisbury Cathedral. The shape 22 m
of the floor plan is called the building’s “footprint.” The student
estimates the dimensions of the cathedral on her sketch at the 46 m 16 m
right. Use the student’s lengths to estimate the area of Salisbury
Cathedral’s footprint.

49. Coordinate Geometry  The endpoints of a diameter of a circle 12 m
are A(2, 1) and B(5, 5). Find the area of the circle in terms of p.

50. Algebra  A rectangle has a base of x units. The area is (4x2 - 2x) 65 m 20 m

square units. What is the height of the rectangle in terms of x?

(4 - x) units (4x3 - 2x2) units

(x - 2) units (4x - 2) units 52 m

Coordinate Geometry  Graph each rectangle in the coordinate 25 m
plane. Find its perimeter and area.

51. A( -3, 2), B( -2, 2), C( -2, -2), D( -3, -2)
52. A( -2, -6), B( -2, -3), C(3, -3), D(3, -6)

53. The surface area of a three-dimensional figure is the sum of the areas of 4HSinM. 11GMSE_0108_a01356
2nd pass 01-06-09
all of its surfaces. You can find the surface area by finding the area of a net Durke 8 in. 6 in.

for the figure.
a. Draw a net for the solid shown. Label the dimensions.
b. What is the area of the net? What is the surface area of the solid?

54. Coordinate Geometry  On graph paper, draw polygon ABCDEFG with vertices

A(1, 1), B(10, 1), C(10, 8), D(7, 5), E(4, 5), F(4, 8), and G(1, 8). Find the perimethersm11gmse_0108_t01220.ai
and the area of the polygon.

66 Chapter 1  Tools of Geometry

55. Pet Care  You want to adopt a puppy from your local animal shelter. First, 1 ft
you plan to build an outdoor playpen along the side of your house, as shown
on the right. You want to lay down special dog grass for the pen’s floor. If dog 6 ft Pen House
grass costs $1.70 per square foot, how much will you spend?
1 ft
56. A rectangular garden has an 8-ft walkway around it. How many more feet is 3 ft
the outer perimeter of the walkway than the perimeter of the garden?

C Challenge Algebra  Find the area of each figure.

57. a rectangle with side lengths 2a units and 3b units
5b 8

58. a square with perimeter 10n units

59. a triangle with base (5x - 2y) units and height (4x + 3y) units

Standardized Test Prep

SAT/ACT 60. An athletic field is a 100 yd-by-40 yd rectangle with a 10 yd
semicircle at each of the short sides. A running track 10 yd
wide surrounds the field. Find the perimeter of the outside 100 yd
of the running track to the nearest tenth of a yard. 40 yd

61. A square garden has a 4-ft walkway around it. The garden
has a perimeter of 260 ft. What is the area of the walkway in
square feet?

62. A(4, -1) and B( -2, 3) are points in a coordinate plane. M is the midpoint of AB.
hsm11gmse_0108_t01221.ai
What is the length of MB to the nearest tenth of a unit?

63. Find CD to the nearest tenth if point C is at (12, -8) and point D is at (5, 19).

Mixed Review

Find (a) AB to the nearest tenth and (b) the midpoint coordinates of AB. See Lesson 1-7.

64. A(4, 1), B(7, 9) 65. A(0, 3), B( -3, 8) 66. A( -1, 1), B( -4, -5)
See Lesson 1-6.
<BG> is the perpendicular bisector of WR at point K.

67. What is m∠BKR?

68. Name two congruent segments.

Get Ready!  To prepare for Lesson 2-1, do Exercise 69. See p. 889.

69. a. Copy and extend this list to show the first 10 perfect squares.
12 = 1, 22 = 4, 32 = 9, 42 = 16, . . .

b. Which do you think describes the square of any odd number?
It is odd. It is even.

Lesson 1-8  Perimeter, Circumference, and Area 67

Concept Byte Comparing Perimeters CMoamthmemonatCicosreFlorida
and Areas StantedSatradnsdards
Use With Lesson 1-8
Prepares for GM-AMFGS.9A1.22 .GA-pMplGy .c1o.n2c eApptsply
technology coof ndceenpstistyobfadseendsiotyn baaresaedanodn vaoreluamaendin
vmooludmeleinign smituoadteiloings. situations.

MP 5

You can use a graphing calculator or spreadsheet software to find maximum and MATHEMATICAL
minimum values. These values help you solve real-world problems where you want
to minimize or maximize a quantity such as cost or time. In this Activity, you will find PRACTICES
minimum and maximum values for area and perimeter problems.



You have 32 yd of fencing. You want to make a rectangular horse pen with
maximum area.

1. Draw some possible rectangular pens and find their areas. Use the examples 11 yd 10 yd
at the right as models.

2. You plan to use all of your fencing. Let X represent the base of the pen. What is

the height of the pen in terms of X? What is the area of the pen in terms of X? 5 yd 6 yd
A ϭ 55 yd2 A ϭ 60 yd2
3. Make a graphing calculator table to find area. Again, let X represent the base.
For Y1, enter the expression you wrote for the height in Question 2. For Y2,
enter the expression you wrote for the area in Question 2. Set the table so that

X starts at 4 and changes by 1. Scroll down the table. X Y1 Y2
a. What value of X gives you the maximum area?
4
hsm11gms56 e_0108b_t01705.ai
b. What is the maximum area? 7

4. Use your calculator to graph Y2. Describe the shape of the graph. Trace on the 8
graph to find the coordinates of the highest point. What is the relationship, if any, 9
10
between the coordinates of the highest point on the graph and your answers to
Y1 ϭ

Question 3? Explain.

Exercises hsm11gmse_0108b_t01168.ai

5. For a fixed perimeter, what rectangular shape will result in a maximum area?

6. Consider that the pen is not limited to polygon shapes. What is the area of a
circular pen with circumference 32 yd? How does this result compare to the
maximum area you found in the Activity?

7. You plan to make a rectangular garden with an area of 900 ft2. You want to use a
minimum amount of fencing to keep the cost low.

a. List some possible dimensions for the garden. Find the perimeter of each.
b. Make a graphing calculator table. Use integer values of the base b, and the

corresponding values of the height h, to find values for P, the perimeter. What
dimensions will give you a garden with the minimum perimeter?

68 Concept Byte  Comparing Perimeters and Areas

1 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in Lessons 1-3,
problems, you 1-4, and 1-5. Use the work you did to complete the following.
will pull together 1. Solve the problem in the Task Description on page 3 by finding the value of each
many concepts
and skills that variable and answering the riddle. Show all your work and explain each step of
you learned in your solution.
this chapter.
They are the 2. Reflect  Choose one of the Mathematical Practices below and explain how you
basic tools applied it in your work on the Performance Task.
used to study
geometry. MP 1: Make sense of problems and persevere in solving them.

MP 3: Construct viable arguments and critique the reasoning of others.

MP 6: Attend to precision.

On Your Own

Cameron found another page from the old riddle book, as shown below.

What flies without wings?

(6t Ϫ 3)Њ
(23m ϩ 4)Њ

62Њ 2e Ϫ 4 Ϫ1 Ϫ e
(9i)Њ

Solve the riddle, before the feast. 69
Arrange the variables from greatest to least.
a. Find the value of each variable.
b. What is the answer to the riddle?

Chapter 1  Pull It All Together

1 Chapter Review

Connecting and Answering the Essential Questions

1 Visualization Lesson 1-1  Nets and Drawings
You can represent a 3-D
figure with a 2-D drawing
by visualizing the surfaces
of the figure and how they Top
relate to each other.
Front Right Front Right

Isometric Orthographic

Lesson 1-2  Points, Lines, and Lesson 1-5  Exploring Angle Pairs

2 Reasoning Phlsamne1s1gmse_0h1scmr_1t011g2m3s3e.a_i01cr_t0Yo1u2c5a9n.acoinclude these relationships from
Geometry is a mathematical an unmarked diagram.
system built on basic Plane Point • Angles are adjacent.
terms, definitions, and Line • Angles are adjacent supplementary.
assumptions called • Angles are vertical angles.
postulates.

Lessons 1-3 and 1-4  Lesson 1-7  Midpoint and Distance

3 Measurement SAegmhe5snmt s1a1BngdmAsAneg_le0s415cЊr_t01261.ai ( )Midpoint =x1+x2,y1 + y2
You can describe the 2 2
attributes of a segment
or angle by using unit AB = 5 m∠A = 45 Distance = 5 1x2 - x122 + 1y2 - y122
amounts.

hsm11gmse_0h1scmr_1t011g2m6s5e.a_i01cr_t01266.ai

Chapter Vocabulary

• acute, right, obtuse, • congruent angles (p. 29) • net (p. 4) • ray, opposite rays (p. 12)
straight angles (p. 29) • congruent segments • orthographic drawing • segment (p. 12)
• segment bisector (p. 22)
• adjacent angles (p. 34) (p. 22) (p. 6) • space (p. 12)
• angle bisector (p. 37) • construction (p. 43) • perpendicular bisector • supplementary angles
• collinear points, coplanar • isometric drawing (p. 5)
• linear pair (p. 36) (p. 44) (p. 34)
(p. 12) • measure of an angle • perpendicular lines (p. 44) • vertex of an angle (p. 27)
• complementary angles • point, line, plane (p. 11) • vertical angles (p. 34)
(p. 28) • postulate, axiom (p. 13)
(p. 34)

Choose the correct term to complete each sentence.
1. A ray that divides an angle into two congruent angles is a(n) ? .
2. ? are two lines that intersect to form right angles.
3. A(n) ? is a two-dimensional diagram that you can fold to form a 3-D figure.
4. ? are two angles with measures that have a sum of 90.

70 Chapter 1  Chapter Review

1-1  Nets and Drawings for Visualizing Geometry

Quick Review Exercises

A net is a two-dimensional pattern that you can fold to form 5. T he net below is for a number cube. What are the
a three-dimensional figure. A net shows all surfaces of a three sums of the numbers on opposite surfaces of
figure in one view. the cube?
An isometric drawing shows a corner view of a three- 5
dimensional object. It allows you to see the top, front, and
side of the object in one view. 1234
An orthographic drawing shows three separate views of
a three-dimensional object: a top view, a front view, and a 6
right-side view.
6. Make an orthographic
Example drawing for the isometric
drawing at the right.
Draw a net for the solid at the right.
Assume there ahresmno11gmse_01cr_t01236.ai

hidden cubes. Front Right

hsm11gmse_01cr_t01234.ai hsm11gmse_01cr_t11742.ai

1-2  Points, Lines, and Planes Exercises

hsm11gmse_01cr_t01235.ai Use the figure below for Exercises 7–9.
TS
Quick Review
Q R
A point indicates a location and has no size. D C
A line is represented by a straight path that extends in two
opposite directions without end and has no thickness. AB
A plane is represented by a flat surface that extends 7. Name two intersecting lines.
without end and has no thickness.
Points that lie on the same line are collinear points.
Points and lines in the same plane are coplanar.
Segments and rays are parts of lines.

Example 8. Name the intersection of planes QRBA and TSRQ.

Name all the segments and rays 9. Namhesmthr1e1egnmonsceo_ll0in1ecarr_pt0oi1n2ts3. 8.ai
in the figure.
D Determine whether the statement is true or false.
Segments: AB, AC, BC, and BD ABC Explain your reasoning.
Rays: BBCA>>,, aCnAd> oBrDC> B>, AC> or AB>,
10. Two points are always collinear.
11. LM> and ML> are the same ray.

hsm11gmse_01cr_t0C1h2ap3t7e.ra1i  Chapter Review 71

1-3  Measuring Segments

Quick Review Exercises

The distance between two points is the length of the For Exercises 12 and 13, use the number line below.
segment connecting those points. Segments with the same PH
length are congruent segments. A midpoint of a segment
divides the segment into two congruent segments. Ϫ4 Ϫ2 0 2 4
12. Find two possible coordinates of Q such that
Example B
PQ = 5.
Are AB and CD congruent?
C AD 13. UthseemthiedpnouimnhtbsoemfrPl1iHn1e.gambosvee_.0F1incdr_thte01co2o4r0d.ianiate of

Ϫ8 Ϫ6 Ϫ4 Ϫ2 0 2 14. Find the value of m.

AB = 0 -3 - 2 0 = 0 -5 0 = 5 3m ϩ 5 4m Ϫ 10
CD = 0 -7 - (-2) 0 = 0 -5 0 = 5
AB = CD, sohAsBm≅11CgDm. se_01cr_t01239.ai ABC

15. If XZ = 50, what are XY and YZ?
a aϩ8

Xhsm11gmY se_01cr_t01Z241.ai

1-4  Measuring Angles hsm11gmse_01cr_t01242.ai

Quick Review Exercises

Two rays with the same endpoint form an angle. The Classify each angle as acute, right, obtuse, or straight.
endpoint is the vertex of the angle. You can classify angles
as acute, right, obtuse, or straight. Angles with the same 16. 17.
measure are congruent angles.
Use the diagram below for Exercises 18 and 19.
Example
hsm11gmse_0N1cr_Mt01245.hasi m11gmse_01cr_t01246.ai
If mjAOB = 47 and mjBOC = 73, find mjAOC.
P
AB QR
18. If m∠MQR = 61 and m∠MQP = 25, find m∠PQR.
OC 19. If m∠NQM = 2x + 8 and m∠PQR = x + 22,
m∠AOC = m∠AOB + m∠BOC
find the value of xh. sm11gmse_01cr_t01247.ai
= 47 + 73

= 120 hsm11gmse_01cr_t01243.ai

72 Chapter 1  Chapter Review

1-5  Exploring Angle Pairs

Quick Review Exercises A B
C
Some pairs of angles have special names. Name a pair of each of the following. D
20. complementary angles
• Adjacent angles: coplanar angles with a common 21. supplementary angles EF
side, a common vertex, and no common interior 22. vertical angles
points 23. linear pair

• Vertical angles: sides are opposite rays Find the value of x.
24. (3x ϩ 31)Њ (2x Ϫ 6)Њ
• Complementary angles: measures have a sum of 90
hsm11gmse_01cr_t012
• Supplementary angles: measures have a sum of 180

• Linear pair: adjacent angles with noncommon sides
as opposite rays

Angles of a linear pair are supplementary.

Example EB 25.
ACD
Are jACE and jBCD vertical h3sxmЊ(41x1Ϫg1m5)sЊ e_01cr_t01250.ai
angles? Explain.

No. They have only one set of
sides with opposite rays.

1-6  Basic Constructionhssm11gmse_01cr_t01248.ai hsm11gmse_01cr_t01251.ai

Quick Review Exercises

Construction is the process of making geometric 26. Use a protractor to draw a 73° angle. Then construct
figures using a compass and a straightedge. Four basic an angle congruent to it.
constructions involve congruent segments, congruent
angles, and bisectors of segments and angles. 27. Use a protractor to draw a 60° angle. Then construct
the bisector of the angle.

Example 28. Sketch LMon paper. Construct a line segment

Construct AB congruent to EF . EF congruent to LM. Then construct the perpendicular
bisector of your line segment.
Step 1
AL M
Draw a ray with endpoint A. 29. a. Sketch ∠B on paper. Construct

Step 2 hsm11gmse_01cr_t01252.ai an angle congruent to ∠B.

Open the compass to the length of hAsm11gmse_B01cr_t012 53 .abi. aCnognlsetrfruohcmtsmthpea1rb1tig(saem)c.tsoer_o0f 1yocur_r t012B55.ai
EF . Keep that compass setting and
put the compass point on point A.
Draw an arc that intersects the ray.
Label the point of intersection B.

hsm11gmse_01cr_t01254.ai hsm11gmse_01cr_t01

Chapter 1  Chapter Review 73

1-7  Midpoint and Distance in the Coordinate Plane

Quick Review Exercises

You can find the coordinates of the midpoint M of Find the distance between the points to the nearest tenth.
30. A( -1, 5), B(0, 4)
AB with endpoints A(x1, y1) and B(x2, y2) using the 31. C( -1, -1), D(6, 2)
Midpoint Formula. 32. E( -7, 0), F(5, 8)

( )M x1 + x2, y1 + y2 AB has endpoints A(− 3, 2) and B(3, − 2).
2 2 33. Find the coordinates of the midpoint of AB.
34. Find AB to the nearest tenth.
You can find the distance d between two points A(x1, y1)
and B(x2, y2) using the Distance Formula. M is the midpoint of JK . Find the coordinates of K.
35. J( -8, 4), M( -1, 1)
d = 5 1x2 - x122 + 1y2 - y122 36. J(9, -5), M(5, -2)
37. J(0, 11), M( -3, 2)
Example

GH has endpoints G(− 11, 6) and H(3, 4). What are the
coordinates of its midpoint M?

x-coordinate = - 11 + 3 = -4
2
6 + 4
y-coordinate = 2 = 5

The coordinates of the midpoint of GH are M( -4, 5).

1-8  Perimeter, Circumference, and Area

Quick Review Exercises

The perimeter P of a polygon is the sum of the lengths of Find the perimeter and area of each figure.
its sides. Circles have a circumference C. The area A of a
polygon or a circle is the number of square units it encloses. 38. 39.
Square:  P = 4s; A = s2
Rectangle:  P = 2b + 2h; A = bh 8 cm
Triangle: P = a + b + c; A = 12bh
Circle:  C = pd or C = 2pr; A = pr2 3 in.

Example 5 in.

Find the perimeter and area of a rectangle with Find the circumference and the area for each
b = 12 m and h = 8 m.
circleh isnm te1r1mgsmofseP_. 01cr_t01257.ai
P = 2b + 2h A = bh
40. r = 3 in. hsm11gmse_01cr_t01258.ai
#= 2(12) + 2(8) = 12 8
41. d = 15 m
= 40 = 96
The perimeter is 40 m and the area is 96 m2.

74 Chapter 1  Chapter Review

1 Chapter Test MathX

OLMathXL® for School
R SCHO Go to PowerGeometry.com
L®

FO

Do you know HOW? Use the figure to complete each statement.
1. Draw a net for a cube.
2. Draw an obtuse ∠ABC. Use a compass and a 11. VW is the ? of AY . V

straightedge to bisect the angle. 12. If EY = 3.5, then AY = ? .

Use the figure for Exercises 3–6. 13. AE = 1 ? A E Y
2 W

14. ? is the midpoint of ? .

A For the given dimensions, find the area of each figure.
If necessary, round to the nearest hundredth.
EB
15. rectangle with base 4 m and heighhstm2 1cm1gmse_01ct_t01182.ai
DQ
C 16. square with side length 3.5 in.

3. Name three collinear points. 17. circle with diameter 9 cm

4. Name four coplanar points. Algebra  Find the value of the variable.

5. What is the intehrssemct1io1ngomf  <sAeC_> a0n1dctp_latn0e11Q7?1.ai BK

6. How many planes contain the given line and point? J
a. <DB> and point A DR
b. <BD> and point E 18. m∠BDK = 3x + 4, m∠JDR = 5x - 10
c. <AC> and point D 19. m∠BDJ = 7y + 2, m∠JDR = 2y + 7
d. <EB> and point C

7. The running track at the right F 212 ft G hsm11gmse_01ct_t01184.ai

is a rectangle with a half circle 100 ft Do you UNDERSTAND?
on each end. FI and GHare Determine whether each statement is always,

diameters. Find the area inside I H sometimes, or never true.

the track to the nearest tenth. 20. LJ> and TJ> are opposite rays.

8. Algebra  M(x, y) is the midpoint of CD with 21. Angles that form a linear pair are supplementary.
endpoints C(5, 9) and D(17, 29).
hsm11gmse_01ct_t011 2725. .Tahi e intersection of two planes is a point.
a. Find the values of x and y.

b. Show that MC = MD. 23. Complementary angles are congruent.

9. Algebra  If JK = 48, find the value of x. 24. Writing  Explain why it is useful to have more than
one way to name an angle.
J HK
25. You have 30 yd2 of carpet. You want to install
4x Ϫ 15 2x ϩ 3 carpeting in a room that is 20 ft long and 15 ft wide.
Do you have enough carpet? Explain.
10. To the nearest tenth, find the perimeter of △ABC

with vertices A1 -2, -22, B(0, 5), and C13, -12.

hsm11gmse_01ct_t01179.ai

Chapter 1  Chapter Test 75

1 Common Core Cumulative ASSESSMENT
Standards Review

Some questions ask you to find What is the distance from the midpoint of TIP 1
a distance using coordinate AB to endpoint B?
geometry. Read the sample The midpoint divides the
question at the right. Then 4y segment into two congruent
follow the tips to answer it. A segments that are each half
of the total length.
TIP 2 Ϫ4 Ϫ2 O x
Ϫ2 246 Think It Through
Use the Distance
Formula to find the Ϫ4 B Find AB using the Distance
length of the segment. Formula.

210 10 AB = 2( - 2 - 6)2 + [3 - ( - 3)]2
5 100
= 2100
= 10
The distance from the midpoint
of AB to endpoint B is 21AB, or 5.
The correct answer is B.

hsm11gmse_01cu_t01225.ai

LVVeooscsacoabnubluarlayry Builder Selected Response

As you solve test items, you must understand Read each question. Then write the letter of the correct
the meanings of mathematical terms. Match each answer on your paper.
term with its mathematical meaning.
1. Points A, B, C, D, and E are collinear. A is to the right of
A. segment I. angles with the same
measure B, E is to the right of D, and B is to the left of C. Which
B . angle
bisector II. a two-dimensional diagram of the following is NOT a possible arrangement of the
of a three-dimensional figure
C. construction points from left to right?
III. the part of a line consisting
D . net of two endpoints and all D, B, A, E, C B, D, E, C, A
points between
E. congruent D, B, A, C, E B, A, E, C, D
angles IV. a ray that divides an angle
into two congruent angles 2. A square and a rectangle have equal area. If the

rectangle is 36 cm by 25 cm, what is the perimeter of

the square?

30 cm 120 cm

V. a geometric figure made 60 cm 900 cm
using a straightedge and
compass

76 Chapter 1  Common Core Cumulative Standards Review

3. Which construction requires drawing only one arc 8. Given: ∠A
with a compass?
What is the second step in constructing the
constructing congruent segments angle bisector of ∠A?

constructing congruent angles B
AD
constructing the perpendicular bisector
C
constructing the angle bisector

4. Rick paints the four walls in a room that is 12 ft long

and 10 ft wide. The ceiling in the room is 8 ft from the

floor. The doorway is 3 ft by 7 ft, and the window is 6 ft

by 5 ft. If Rick does NOT paint the doorway or window, Draw AD>.

what is the approximate area that he paints? From poinhtssmB1a1ndgmC,sues_e0th1ecusa_mt0e1c2om27p.aasis setting

301 ft2 331 ft2 to draw arcs that intersect at D.

322 ft2 352 ft2 Draw a line segment connecting points B and C.

5. If ∠A and ∠B are supplementary angles, what angle From point A, draw an arc that intersects the sides
relationship between ∠A and ∠B CANNOT be true? of the angle at points B and C.

∠A and ∠B are right angles. 9. What is the distance, to the nearest tenth, from point D
to point E through points C, A, B, and F ?
∠A and ∠B are adjacent angles.
y
∠A and ∠B are complementary angles.
4
∠A and ∠B are congruent angles. E
A2
6. A net for a small rectangular gift box is shown below. D F
What is the total area of the net? Ϫ2 C O
Ϫ2 B x
9 cm 24 6
13 cm

18 cm

14.0 28.2

468 cm2 1026 cm2 18.7 hsm11gmse_01cu _3t40.41228.ai

782 cm2 2106 cm2 1 0. Which postulate most closely resembles the Angle
Addition Postulate?
7. The mehassmur1e1ogf amnsaen_g0le1icsu1_2tle0s1s2th2a6n.atiwice the
Ruler Postulate
measure of its supplement. What is the measure of
the angle? Protractor Postulate

28 64 Segment Addition Postulate

34 116 Area Addition Postulate

11. What is the length of the segment with endpoints
A(1, 7) and B( -3, -1)?

240 280

8 40

Chapter 1  Common Core Cumulative Standards Review 77

Constructed Response 2 3. Copy the graph below. Connect the midpoints of
the sides of the square consecutively. What is the
12. The measure of an angle is 78 less than the measure of perimeter of the new square? Show your work.
its complement. What is the measure of the angle?
y
1 3. The face of a circular game token has an area of
10p cm2. What is the diameter of the game token? 4 x
Round to the nearest hundredth of a centimeter. 2 4

1 4. The measure of an angle is one third the measure of its Ϫ6 Ϫ4 Ϫ2 O
supplement. What is the measure of the angle? Ϫ2
Ϫ4
15. Bill’s bike wheels have a 26-in. diameter. The odometer
on his bike counts the number of times a wheel rotates
during a trip. If the odometer counts 200 rotations
during the trip from Bill’s house to school, how many
feet does Bill travel? Round to the nearest foot.

1 6. Y is the midpoint of XZ. What is the value of b?

2b Ϫ 1 26 Ϫ 4b Extended Response

XYZ 24. Make an ohrsthmo1gr1agpmhicsed_ra0w1icnug_otf0th1e2t3h1re.aei-

17. A rectangular garden has dimensions 6 ft by 17 ft. A dimensional drawing. How might an orthographic
second rectangular garden has dimensions 4 yd by drawing of the figure be more useful than a net?

t9hyedt.whWoshmgaa1trid1segtnhmse?saere_a0i1ncsuq_uta0re12fe2e9t o.afithe larger of RIGHTFRONT

1 8. The sum of the measures of a complement and a 25. A packaging company wants to fit 6 energy-drink cans
supplement of an angle is 200. What is the measure of in a cardboard box, as shown below. The bottom of
the angle? each can is a cgiercolme 1w2it_hsea_nccasr_eca0o1f_9cpsr.cami 2.

19. What is the area in square units of a rectangle with a. What is the total area taken
vertices ( -2, 5), (3, 5), (3, -1), and ( -2, -1)? up by the bottoms of the
cans? Round to the nearest
2 0. AB has endpoints A( -4, 5) and B(3, 5). What is the hundredth.
x‑coordinate of a point C such that B is the midpoint
of AC? b. Will the cans fit in a box
with length 16 cm and height 12 cm? Explain.
21. The two blocks of cheese shown below are cut into
slices of equal thickness. If the cheese sells at the same
cost per slice, which type of cheese slice is the better
deal? Explain your reasoning.

Provolone American

10 cm 9 cm
9 cm

2 2. The midpoint of GI is (2, -1). One endpoint is hsm11gmse_01cu_t01232.ai
G( -1, -3). What are the coordinates of endpoint I ?

hsm11gmse_01cu_t01230.ai

78 Chapter 1  Common Core Cumulative Standards Review

CHAPTER

Get Ready! 2

S kills Evaluating Expressions

Handbook, Algebra   Evaluate each expression for the given value of x.
p. 882

1. 9x - 13 for x = 7 2. 90 - 3x for x = 31 3. 21x + 14 for x = 23

Skills Solving Equations

Handbook, Algebra  Solve each equation.
p. 886

4. 2x - 17 = 4 5. 3x + 8 = 53

6. (10x + 5) + (6x - 1) = 180 7. 14x = 2(5x + 14)

8. 2(x + 4) = x + 13 9. 7x + 5 = 5x + 17

10. (x + 21) + (2x + 9) = 90 11. 2(3x - 4) + 10 = 5(x + 4)

Lessons 1-3 Segments and Angles A

through 1-5 Use the figure at the right.

12. Name ∠1 in two other ways. xϩ8

13. If D is the midpoint of AB, find the value of x. D

14. If ∠ACB is a right angle, m∠1 = 4y, and m∠2 = 2y + 18, 12 2x ϩ 5
find m∠1 and m∠2. C B

15. Name a pair of angles that form a linear pair.

16. Name a pair of adjacent angles that are not supplementary.

17. If m∠ADC + m∠BDC = 180, name the straight angle. hsm11gmse_02co_t00578

Looking Ahead Vocabulary

18. A scientist often makes an assumption, or hypothesis, about a scientific problem.
Then the scientist uses experiments to test the hypothesis to see if it is true. How
might a hypothesis in geometry be similar? How might it be different?

19. The conclusion of a novel answers questions raised by the story. How do you think
the term conclusion applies in geometry?

20. A detective uses deductive reasoning to solve a case by gathering, combining, and
analyzing clues. How might you use deductive reasoning in geometry?

Chapter 2  Reasoning and Proof 79

2CHAPTER Reasoning and Proof

Download videos VIDEO Chapter Preview Reasoning and Proof
connecting math
to your world.. 2-1 Patterns and Inductive Reasoning Essential Question  How can you make
2-2 Conditional Statements a conjecture and prove that it is true?
2-3 Biconditionals and Definitions
Interactive! YNAM IC 2-4 Deductive Reasoning
Vary numbers, T I V I TIAC 2-5 Reasoning in Algebra and Geometry
graphs, and figures D 2-6 Proving Angles Congruent
to explore math ES
concepts..

The online
Solve It will get
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Construct Viable Arguments
Spanish English/Spanish Vocabulary Audio Online: • Attend to precision

English Spanish

biconditional, p. 98 bicondicional

Online access conclusion, p. 89 conclusión
to stepped-out
problems aligned conditional, p. 89 condicional
to Common Core
Get and view conjecture, p. 83 conjetura
your assignments
online. NLINE contrapositive, p. 91 contrapositivo
ME WO
O converse, p. 91 recíproco
RK
HO deductive reasoning, p. 106 razonamiento deductivo

hypothesis, p. 89 hipótesis

inductive reasoning, p. 82 razonamiento inductivo

Extra practice inverse, p. 91 inverso
and review
online negation, p. 91 negación

theorem, p. 120 teorema

Virtual NerdTM
tutorials with
built-in support

PERFORRMMAANNCCTASK
E
Common Core Performance Task

Analyzing a Calendar Pattern

YYoouu ccaann cchhoooossee aa sseett off ffoouurr nnuummbbeerrss ffrroomm aa ppaaggee ooff aa mmoonntthhllyy ccaalleennddaarr bbyy
ddrraawwiinngg aa 22--bbyy--22 ssqquuaarree aarroouunndd ffoouurr ddaatteess,, aass sshhoowwnn bbeellooww.. TThhee ffoouurr nnuummbbeerrss
ffoorrmm aa 22--bbyy--22 ccaalleennddaarr ssqquuaarree..

MARCH

SSUUNN MMOONN TTUUEE WWEEDD TTHHUU FFRRII SSAATT
11 22 33

44 55 66 77 88 99 1100

1111 1122 1133 1144 1155 1166 1177

1188 1199 2200 2211 2222 2233 2244

2255 2266 2277 2288 2299 3300 3311

Task Description

UUssee iinndduuccttiivvee rreeaassoonniinngg ttoo mmaakkee aa ccoonnjjeeccttuurree aabboouutt tthhee nnuummbbeerrss oonn tthhee
ddiiaaggoonnaallss ooff aa 22--bbyy--22 ccaalleennddaarr ssqquuaarree.. TThheenn uussee ddeedduuccttiivvee rreeaassoonniinngg ttoo pprroovvee
yyoouurr ccoonnjjeeccttuurree..

Connecting the Task to the Math Practices MMAATTHHEEMMAATTIICCAALL

AAss yyoouu ccoommpplleettee tthhee ttaasskk,, yyoouu’’llll aappppllyy sseevveerraall SSttaannddaarrddss ffoorr MMaatthheemmaattiiccaall PRACTICES
PPrraaccttiiccee..

•• YYoouu’’llll iiddeennttiiffyy aa ppaatttteerrnn iinn 22--bbyy--22 ccaalleennddaarr ssqquuaarreess aanndd rreeaassoonn iinndduuccttiivveellyy ttoo
mmaakkee aa ccoonnjjeeccttuurree.. ((MMPP 33))

•• YYoouu’’llll aatttteenndd ttoo pprreeciissiioonn aass yyoouu ccoonnssiiddeerr hhooww ttoo wwrriittee yyoouurr ccoonnjjeeccttuurree aanndd
rreellaatteedd sseenntteenncceess aass ccoonnddiittiioonnaall ssttaatteemmeennttss.. ((MMPP 66))

•• YYoouu’’llll llooookk ffoorr ssttrruuccttuurree iinn tthhee nnuummbbeerrss oonn tthhee ccaalleennddaarr.. ((MMPP 77))

Chapter 2  Reasoning and Proof 81

2-1 Patterns and Inductive CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Reasoning
Prepares for GM-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosvaebout lines
tahnedoarenmglsesa.bAolustolinPersepanadreasngfolers.GA-lCsOo.PCr.1e0p,aGre-CsOfo.Cr.11
MMAPF1S,.M91P2.3G, -MCOP.43,.1M0P, M7AFS.912.G-CO.3.11
MP 1, MP 3, MP 4, MP 7

Objective To use inductive reasoning to make conjectures

See if you can Fold a piece of paper in half. When you unfold it,
find a pattern to the paper is divided into two rectangles. Refold
help you solve this the paper, and then fold it in half again. This time
problem. when you unfold it, there are four rectangles. How
many rectangles would you get if you folded a piece
of paper in half eight times? Explain.

MATHEMATICAL In the Solve It, you may have used inductive reasoning. Inductive reasoning is

PRACTICES reasoning based on patterns you observe.

Essential Understanding  You can observe patterns in some number sequences
and some sequences of geometric figures to discover relationships.

Problem 1 Finding and Using a Pattern

How do you look Look for a pattern. What are the next two terms in each sequence?

f or a pattern in a A 3, 9, 27, 81, . . . B
Ls oeoqkufeonr caer?elationship
3 9 27 81

between terms. Test ϫ3 ϫ3 ϫ3

tchoa ntsitshteenret ltahtriooungshhiopuits Each term is three times the previous Each circle contains a polygon that has

the sequence. term. The next two terms are one more side than the preceding

81 * 3 = 243 and 243 * 3 = 729. sphioxs‑lmysgid1oe1ndg. Tamhnedsena_es0xet2vte0wn1o‑_scitid0rec0dle6sp0oc7oly.nagtioanin. a

h sm11gmse_0201_ t00606.ai

Less on

Vocabulary
• inductive

reasoning

•• ccoounnje tcetruerxeample Got It? 1. Wa. h4a5t,a4r0e,t3h5e, 3n0e,xtctwo terms in each sequben. ce?

hsm11gmse_0201_t00608.ai

82 Chapter 2  Reasoning and Proof hsm11gmse_0201_t00609.ai

Do you need to You may want to find the tenth or the one-hundreth term in a sequence. In this case,
draw a circle with rather than find every previous term, you can look for a pattern and make a conjecture.
20 diameters? A conjecture is a conclusion you reach using inductive reasoning.
No. Solve a simpler
problem by finding Problem 2 Using Inductive Reasoning
the number of regions Look at the circles. What conjecture can you make
formed by 1, 2, and about the number of regions 20 diameters form?
3 diameters. Then look
for a pattern. 1 diameter forms 2 regions.

hsm11gmse_0201_t00610.ai

2 diameters form 4 regions.

3 diameters form 6 regions.

#Each circle has twice as many regions as diameters. Twenty diameters form

20 2, or 40 regions.

Got It? 2. What conjecture can you make about the twenty-first term in
R, W, B, R, W, B, c ?

It is important to gather enough data before you make a conjecture. For example, you

#do not have enough information about the sequence 1, 3, c to make a reasonable

conjecture. The next term could be 3 3 = 9 or 3 + 2 = 5.

What’s the first step? Problem 3 Collecting Information to Make a Conjecture
Start by gathering data. What conjecture can you make about the sum of the first 30 even numbers?
You can organize your Find the first few sums and look for a pattern.
data by making a table.
Number of Terms Sum
1
2 2 ϭ 2ϭ1и2
3
4 2ϩ4 ϭ 6ϭ2и3 Each sum is the product of the
number of terms and the number
2 ϩ 4 ϩ 6 ϭ 12 ϭ 3 и 4 of terms plus one.

2 ϩ 4 ϩ 6 ϩ 8 ϭ 20 ϭ 4 и 5

#You can conclude that the sum of the first 30 even numbers is 30 31, or 930.

Got It? 3. What conjecture can you make about the sum of the first 30 odd numbers?
hsm11gmse_0201_t00612

Lesson 2-1  Patterns and Inductive Reasoning 83

Problem 4 Making a Prediction

How can you use the Sales  Sales of backpacks at a nationwide company Backpacks Sold

given data to make a decreased over a period of six consecutive months. What Number 11,000
prediction? conjecture can you make about the number of backpacks 10,500
Look for a pattern of the company will sell in May? 10,000
points on the graph. Then
make a prediction, based The points seem to fall on a line. The graph shows the 9500
on the pattern, about number of sales decreasing by about 500 backpacks each 9000
where the next point month. By inductive reasoning, you can estimate that the 8500
will be. company will sell approximately 8000 backpacks in May. 8000

0
ND J FMAM

Got It? 4. a. What conjecture can you make about Month

backpack sales in June?

b. Reasoning  Is it reasonable to use this graph to make a conjecture about

sales in August? Explain. hsm11gmse_0201_t00614

Not all conjectures turn out to be true. You should test your conjecture multiple
times. You can prove that a conjecture is false by finding one counterexample. A
counterexample is an example that shows that a conjecture is incorrect.

Problem 5 Finding a Counterexample
What is a counterexample for each conjecture?
A If the name of a month starts with the letter J, it is a summer month.
Counterexample: January starts with J and it is a winter month.
B You can connect any three points to form a triangle.
Counterexample: If the three points lie on a line, you cannot form a triangle.

What numbers should These three points . . . but these three points are a
support the conjecture . . . counterexample to the conjecture.

yo u guess-and-check? C When you multiply a number by 2, the product is greater than
Try positive numbers, the original number.
negative numbers,
#cfraa scetsiolnikse, aznedros.pecial Tahnhsedm czoe1nr1oje.gcmtusree_is0t2ru0e1f_otr0p0o6si1ti5ve numbers, but it is false for negative numbers
Counterexample: -4 2 = -8 and -8 w -4.

Got It? 5. What is a counterexample for each conjecture?
a. If a flower is red, it is a rose.
b. One and only one plane exists through any three points.
c. When you multiply a number by 3, the product is divisible by 6.

84 Chapter 2  Reasoning and Proof

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
What are the next two terms in each sequence?
1. 7, 13, 19, 25, c 4. Vocabulary  How does the word counter help you
2. understand the term counterexample?

3. What is a counterexample for the following conjecture? 5. Compare and Contrast  Clay thinks the next term in
All four-sided figures are squares. the sequence 2, 4, c is 6. Given the same pattern,
Ott thinks the next term is 8, and Stacie thinks the
hsm11gmse_0201_t00617 next term is 7. What conjecture is each person
making? Is there enough information to decide who
is correct?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find a pattern for each sequence. Use the pattern to show the next two terms. See Problem 1.

6. 5, 10, 20, 40, c 7. 1, 4, 9, 16, 25, c 8. 1, -1, 2, -2, 3, c

9. 1, 1 , 1 , 81, c 10. 1, 1 , 31, 14, c 11. 15, 12, 9, 6, c
2 4 2 14. 1, 2, 6, 24, 120, c

12. O, T, T, F, F, S, S, E, c 13. J, F, M, A, M, c

15. Washington, Adams, Jefferson, c 16. dollar coin, half dollar, quarter, c

17. AL, AK, AZ, AR, CA, c 18. Aquarius, Pisces, Aries, Taurus, c

19. 20.

Use the sequence and inductive reasoning to make a conjecture. See Problem 2.

hsm11gmse_0201_t11572.ai hsm11gmse_0201_t00621

21. What is the color of the fifteenth figure? 22. What is the shape of the twelfth figure?

23. What is the color of the thirtieth figure? 24. What is the shape of the fortieth figure?

Makehasmco1n1jegcmturseef_o0r2e0a1ch_ts0ce0n6a2r4io. Show your work. See Problem 3.

25. the sum of the first 100 positive odd numbers 26. the sum of the first 100 positive even numbers

27. the sum of two odd numbers 28. the sum of an even and odd number

29. the product of two even numbers 30. the product of two odd numbers

Lesson 2-1  Patterns and Inductive Reasoning 85

STEM Weather  Use inductive reasoning to make a prediction about the weather. See Problem 4.

31. Lightning travels much faster than thunder, 32. The speed at which a cricket chirps

so you see lightning before you hear thunder. is affected by the temperature. If you

If you count 5 s between the lightning hear 20 cricket chirps in 14 s, what

and thunder, how far away is the storm? is the temperature?

Distance of Storm (mi) Number of Chirps
6 per 14 Seconds Temperature (؇F)
5 45
4 10 55
2 15 65

0 0 10 20 30 40
Seconds Between

Lightning and Thunder

Find one counterexample to show that each conjecture ishfaslmse1. 1gmse_0201_t00631 See Problem 5.

33. ∠1 and ∠2 are supplementary, so one of the angles is acute.

34. △hAsBmC1i1sgamrigshet_t0ri2an0g1l_e,ts0o0∠62A6measures 90.

35. The sum of two numbers is greater than either number.

36. The product of two positive numbers is greater than either number.

37. The difference of two integers is less than either integer.

B Apply Find a pattern for each sequence. Use inductive reasoning to show the next
two terms.

38. 1, 3, 7, 13, 21, c 39. 1, 2, 5, 6, 9, c 40. 0.1, 0.01, 0.001, c

41. 2, 6, 7, 21, 22, 66, 67, c 42. 1, 3, 7, 15, 31, c 43. 0, 12, 3 , 87, 1165, c
4

Predict the next term in each sequence. Use your calculator to verify your answer.

44. 12345679 * 9 = 111111111 45. 11111 * 11111 = 1
12345679 * 18 = 222222222 11111 * 11111 = 121
12345679 * 27 = 333333333 11111 * 11111 = 12321

12345679 * 36 = 444444444 11111 * 11111 = 1234321
12345679 * 45 = ■ 11111 * 11111 = ■

46. Patterns  Draw the next figure in the
sequence. Make sure you think about
color and shape.

hsm11gmse_0201_t00629

86 Chapter 2  Reasoning and Proof

Draw the next figure in each sequence. 48.
47.

49. Reasoning  Find the perimeter when 100 triangles are put together in the pattern
shown. Assume that all triangle sides are 1 cm long.

50. Thhsinmk1A1bgomutsae_Pl0a2n0  1B_elto0w06ar2e715 points. Most of the phoisnmts1fi1t gampasttee_rn0.2W01h_icth00628

does not? Explain.

A(6, -2) B(6, 5) C(8, 0) D(8, 7) E(10, 2) F(10, 6) G(11, 4) H(12, 3) hsm11gmse_0201_t006
I(4, 0) J(7, 6) K(5, 6) L(4, 7) M(2, 2) N(1, 4) O(2, 6)

• How can you draw a diagram to help you find a pattern?
• What pattern do the majority of the points fit?

51. Language  Look for a pattern in the Chinese Chinese Number System
number system.
Number Chinese Number Chinese
a. What is the Chinese name for the Word Word
numbers 43, 67, and 84? 1 9
2 y¯ı 10 j˘ıu
b. Reasoning  Do you think that the Chinese 3 11
number system is base 10? Explain. 4 èr 12 shí
5 Ӈ shí-yı¯
52. Open-Ended  Write two different number 6 sa¯n 20 shí-èr
sequences that begin with the same two 7 21
numbers. 8 sì Ӈ Ӈ
30 èr-shí
53. Error Analysis  For each of the past four wu˘ èr-shí-yı¯
years, Paulo has grown 2 in. every year. He is
now 16 years old and is 5 ft 10 in. tall. He lìu Ӈ
figures that when he is 22 years old he will sa¯ n-shí
be 6 ft 10 in. tall. What would you tell Paulo q¯ı
about his conjecture? ba¯

STEM 54. Bird Migration  During bird migration, volunteers get up Bird Count

aoearberslepyroovsneteiBndirtodhneDliirnacyeottomo rhmeeculopnrisdtcytihedneutrniisuntsmgaabne2dr4s-ohtfubpdieerrdniotssdpt.erRcahiceeksssumtthlht1ese1ygmse_Y0e2a0r1_to0Nf0uS6pme3bc7eiers
2004 70
migration.

a. Make a graph of the data. 2005 83

b. Use the graph and inductive reasoning to make a 2006 80
conjecture about the number of bird species the 2007 85
volunteers in this community will observe in 2015.

55. Writing  Describe a real-life situation in which you recently 2008 90

used inductive reasoning.

hsm11gmse_0201_t00638

Lesson 2-1  Patterns and Inductive Reasoning 87

C Challenge 56. History  When he was in the third grade, German mathematician Karl Gauss
(1777–1855) took ten seconds to sum the integers from 1 to 100. Now it’s your turn.
Find a fast way to sum the integers from 1 to 100. Find a fast way to sum the integers
from 1 to n. (Hint: Use patterns.)

57. Chess  The small squares on a chessboard can be combined to form larger squares.

For example, there are sixty-four 1 * 1 squares and one 8 * 8 square. Use inductive
reasoning to determine how many 2 * 2 squares, 3 * 3 squares, and so on, are on a
chessboard. What is the total number of squares on a chessboard?

58. a. Algebra  Write the first six terms of the sequence that starts with 1, and for 2n ϩ 1
which the difference between consecutive terms is first 2, and then 3, 4, 5, and 6. n

b. Evaluate n2 2+ n for n = 1, 2, 3, 4, 5, and 6. Compare the sequence you get with
your answer for part (a).

c. Examine the diagram at the right and explain how it illustrates a value of n2 + n.
PERFO 2

2n2+n=
d. Draw a similar diagram to represent 2 for n 5.

E TASKRMANC PMRATAHCEMTAITChICEsASmL 11gmse_0201_t00640.

Apply What You’ve Learned MP 3

Look back at the information about 2-by-2 calendar squares on page 81.
The March calendar from page 81 is shown again below.

MARCH

SUN MON TUE WED THU FRI SAT
123

4 5 6 7 8 9 10

11 12 13 14 15 16 17

18 19 20 21 22 23 24

25 26 27 28 29 30 31

a. Use the 2-by-2 calendar square outlined in red on the March calendar. Find a
relationship between the pairs of numbers on the diagonals of the calendar square.

b. Repeat part (a) for several other 2-by-2 calendar squares on the March calendar.
Make a conjecture based on your results.

c. Test your conjecture using other calendar months. Can you find a counterexample
to your conjecture?

88 Chapter 2  Reasoning and Proof

2-2 Conditional Statements CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Prepares for GM-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosvaebout lines
tahnedoarenmglsesa.bAolustolinPersepanadreasngfolers.GA-lCsOo.PCr.1e0p,aGre-CsOfo.Cr.11
MMAPF3S,.M91P2.6G, -MCOP.73.10, MAFS.912.G-CO.3.11
MP 3, MP 6, MP 7

Objectives To recognize conditional statements and their parts
To write converses, inverses, and contrapositives of conditionals

The company that prints the bumper sticker at the left below
accidentally reworded the original statement and printed the sticker
three different ways. Suppose the original bumper sticker is true. Are
the other bumper stickers true or false? Explain.

Try to make sense
of the statements
on the bumper
stickers by
figuring out when
they are true, and
when they are
false.

MATHEMATICAL

PRACTICES

The study of if-then statements and their truth values is a foundation of reasoning.

Essential Understanding  You can describe some mathematical relationships
using a variety of if-then statements.

Lesson Key Concept  Conditional Statements

Vocabulary Definition Symbols Diagram
• conditional A conditional is an if-then statement. pSq q
• hypothesis
• conclusion The hypothesis is the part p following if. Read as p
• truth value “if p then q” or
• negation The conclusion is the part q following then. “p implies q.”
• converse
• inverse
• contrapositive
• equivalent

statements

The Venn diagram above illustrates how the set of things that satisfy the hypothehssismli1es1gmse_0202_t006
inside the set of things that satisfy the conclusion.

Lesson 2-2  Conditional Statements 89

Problem 1 Identifying the Hypothesis and the Conclusion

What would a Venn What are the hypothesis and the conclusion of the conditional?
diagram look like? If an animal is a robin, then the animal is a bird.
A robin is a kind of bird,
so the set of robins (R) Hypothesis (p): An animal is a robin.
should be inside the set Conclusion (q): The animal is a bird.
of birds (B).

B Got It? 1. What are the hypothesis and the conclusion of the conditional?
R If an angle measures 130, then the angle is obtuse.

Problem 2 Writing a Conditional
Which part hofsmthe11gmse_02H0o2w_ct0an06yo4u3.wari ite the following statement as a conditional?
Vertical angles share a vertex.
statement is the

hypothesis (p)? Step 1 Identify the hypothesis and the conclusion.
For two angles to be

vertical, they must share Vertical angles share a vertex.
a vertex. So the set of

vertical angles (p) is Step 2 Write the conditional.

inside the set of angles If two angles are vertical, then they share a vertex.
that share a vertex (q).

Got It? 2. How can you write “Dolphins are mammals” as a conditional?

The truth value of a conditional is either true or false. To show that a conditional
is true, show that every time the hypothesis is true, the conclusion is also true. A
counterexample can help you determine whether a conditional with a true hypothesis
is true. To show that the conditional is false, if you find one counterexample for which
the hypothesis is true and the conclusion is false, then the truth value of the conditional
is false.

Problem 3 Finding the Truth Value of a Conditional

Is the conditional true or false? If it is false, find a counterexample.

A If a woman is Hungarian, then she is European.

The conditional is true. Hungary is a European nation, so Hungarians are

How do you find a European.

c ounterexample? B If a number is divisible by 3, then it is odd.
Find an example where
the hypothesis is true, The conditional is false. The number 12 is divisible by 3, but it is not odd.

but the conclusion is Got It? 3. Is the conditional true or false? If it is false, find a counterexample.
f alse. For part (B), find
a  number divisible by 3 a. If a month has 28 days, then it is February.
t hat is not odd. b. If two angles form a linear pair, then they are supplementary.

90 Chapter 2  Reasoning and Proof

The negation of a statement p is the opposite of the statement. The symbol is ∼p and is
read “not p.” The negation of the statement “The sky is blue” is “The sky is not blue.” You
can use negations to write statements related to a conditional. Every conditional has
three related conditional statements.

Key Concept  Related Conditional Statements

Statement How to Example Symbols How to
Conditional Write It pSq Read It
Converse If m∠A = 15, qSp If p, then q.
Inverse Use the given then ∠A is ∼p S ∼q
hypothesis and acute. If q, then p.
Contrapositive conclusion. ∼q S ∼p
If ∠A is If not p,
Exchange the acute, then then not q.
hypothesis and m∠A = 15.
the conclusion. If m∠A ≠ 15, If not q,
then ∠A is not then not p.
Negate both the acute.
hypothesis and
the conclusion of If ∠A is not
the conditional. acute, then
m∠A ≠ 15.
Negate both the
hypothesis and
the conclusion
of the converse.

Below are the truth values of the related statements above. Equivalent statements have
the same truth value.

Statement Example Truth Value
Conditional If m∠A = 15, then ∠A is acute. True
Converse If ∠A is acute, then m∠A = 15. False
Inverse If m∠A ≠ 15, then ∠A is not acute. False
Contrapositive If ∠A is not acute, then m∠A ≠ 15. True

A conditional and its contrapositive are equivalent statements. They are either both true
or both false. The converse and inverse of a statement are also equivalent statements.

Lesson 2-2  Conditional Statements 91

Problem 4 Writing and Finding Truth Values of Statements

What are the converse, inverse, and contrapositive of the following conditional?
What are the truth values of each? If a statement is false, give a counterexample.
If the figure is a square, then the figure is a quadrilateral.

Identify the hypothesis and p: The figure is a square.
the conclusion.
q: The figure is a quadrilateral.
To write the converse, switch
the hypothesis and the Converse: If the figure is a
conclusion. Write q S p. quadrilateral, then the figure is
a square.
To write the inverse, negate
both the hypothesis and the The converse is false.
conclusion of the conditional. Counterexample:
Write ∼ p S ∼ q. A rectangle that is not a square.

Inverse: If the figure is not a
square, then the figure is not a
quadrilateral.

The inverse is false.
Counterexamples:

To write the contrapositive, Contrapositive: If the figure
negate both the hypothesis is not a quadrilateral, then the
and the conclusion of the figure is not a square.
converse. Write ∼ q S ∼ p.
Thhsemc1on1tgrmapossei_ti0ve20is2t_rtu0e0. 644.ai

Got It? 4. What are the converse, inverse, and contrapositive of the conditional
statement below? What are the truth values of each? If a statement is false,

give a counterexample.
If a vegetable is a carrot, then it contains beta carotene.

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. What are the hypothesis and the conclusion of the
3. Error Analysis  Your classmate rewrote the statement
following statement? Write it as a conditional.
Residents of Key West live in Florida. “You jog every Sunday” as the following conditional.

2. What are the converse, inverse, and contrapositive of What is your classmate’s error? Correct it.
the statement? Which statements are true? If you jog, then it is Sunday.

If a figure is a rectangle with sides 2 cm and 3 cm, 4. Reasoning  Suppose a conditional statement and its
then it has a perimeter of 10 cm. converse are both true. What are the truth values of
the contrapositive and inverse? How do you know?

92 Chapter 2  Reasoning and Proof

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Identify the hypothesis and conclusion of each conditional. See Problem 1.

5. If you are an American citizen, then you have the right to vote.

6. If a figure is a rectangle, then it has four sides.

7. If you want to be healthy, then you should eat vegetables.

Write each sentence as a conditional. See Problem 2.
8. Hank Aaron broke Babe Ruth’s home-run record.
9. Algebra  3x - 7 = 14 implies that 3x = 21.
10. Thanksgiving in the United States falls on the fourth Thursday of November.
11. A counterexample shows that a conjecture is false.
12. Coordinate Geometry  A point in the first quadrant has two positive coordinates.

Write a conditional statement that each Venn diagram illustrates.

13. 14. Integers 15. Grains
Colors Wheat
Whole
Blue numbers

Determine if the conditional is true or false. If it is false, find a counterexample. See Problem 3.

16. If a polygon has eight sides, then it is an octagon.

17. Ihf ysomu1li1vge minsaec_o0u2n0tr2y_tht0at0b6o4r5d.earis thhesmUn1it1egdmStsaete_s0, t2h0e2n_yto0u0l6iv4e6in.aCi anhasdma.11gmse_0202_t00647.ai

18. If you play a sport with a ball and a bat, then you play baseball.

19. If an angle measures 80, then it is acute.

If the given statement is not in if-then form, rewrite it. Write the converse, See Problem 4.
inverse, and contrapositive of the given conditional statement. Determine the
truth value of all four statements. If a statement is false, give a counterexample.

20. If you are a quarterback, then you play football.

21. Pianists are musicians.

22. Algebra  If 4x + 8 = 28, then x = 5.
23. Odd natural numbers less than 8 are prime.

24. Two lines that lie in the same plane are coplanar.

Lesson 2-2  Conditional Statements 93

B Apply Write each statement as a conditional.

25. “We’re half the people; we should be half the Congress.” —Jeanette Rankin,
former U.S. congresswoman, calling for more women in office

26. “Anyone who has never made a mistake has never tried anything new.”
—Albert Einstein

27. Probability  An event with probability 1 is certain to occur.

28. Think About a Plan  Your classmate claims that the conditional and contrapositive

of the following statement are both true. Is he correct? Explain.
If x = 2, then x2 = 4.
• Can you find a counterexample of the conditional?
• Do you need to find a counterexample of the contrapositive to know its

truth value?

29. Open-Ended  Write a true conditional that has a true converse, and write a true
conditional that has a false converse.

30. Multiple Representations  Write three separate conditional statements Athletes
that the Venn diagram illustrates. Baseball
players
31. Error Analysis  A given conditional is true. Natalie claims its
contrapositive is also true. Sean claims its contrapositive is false. Who is Pitchers
correct and how do you know?

Draw a Venn diagram to illustrate each statement.

32. If an angle measures 100, then it is obtuse.

33. If you are the captain of your team, then you are a junior or senior.

34. Peace Corps volunteers want to help other people. hsm11gmse_0202_t00648.ai

Algebra  Write the converse of each statement. If the converse is true, write
true. If it is not true, provide a counterexample.

35. If x = -6, then 0 x 0 = 6. 36. If y is negative, then -y is positive.

37. If x 6 0, then x3 6 0. 38. If x 6 0, then x2 7 0.

39. Advertising  Advertisements often suggest conditional
statements. What conditional does the ad at the right imply?

Write each postulate as a conditional statement.
40. Two intersecting lines meet in exactly one point.
41. Two congruent figures have equal areas.
42. Through any two points there is exactly one line.

94 Chapter 2  Reasoning and Proof

C Challenge Write a statement beginning with all, some, or no to match each Venn diagram.

43. Integers 44. 45.
divisible by 2 Triangles Students

Integers Squares Musicians
divisible

by 8

46. Let a represent an integer. Consider the five statements r, s, t, u, and v.

r: a is even. s: a is odd. t: 2a is even. u: 2a is odd. v: 2a + 1 is odd.

Hhoswmm1a1ngymstsaete_m0e2n0t2s_otf0th0e6f4o9rm.aip Shsqmc1a1ngymousme_a0ke2f0r2om_t0th0e6se50st.aatiemhensmts?11gmse_0202_t00651.ai

Decide which are true, and provide a counterexample if they are false.

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 6

In the Apply What You’ve Learned in Lesson 2-1, you made a conjecture based

on patterns you observed in the numbers on the calendar page shown on page 81.

Your conjecture may have been similar to the one below.

Conjecture: The sums of the numbers on the diagonals of a 2-by-2 calendar
square are equal.

You can express this conjecture with symbols by using a, b, c, and d to represent
any four distinct numbers from a monthly calendar page. Assume the list a, b, c, d
gives the numbers in order from least to greatest.

a. Which of the following conditional statements represents the conjecture? What is
the hypothesis of the conditional? What is the conclusion?

I. If a + d = b + c, then a, b, c, and d form a 2-by-2 calendar square.
II. If a, b, c, and d form a 2-by-2 calendar square, then a + d = b + c.

b. Identify the other statement in part (a) as the converse, inverse, or contrapositive
of the conjecture. Then write the other two of these (converse, inverse, or
contrapositive) in if-then form.

c. You now have four conditional statements about distinct numbers a, b, c, and
d from a monthly calendar page. For which of these statements can you find
counterexamples? If possible, give at least two counterexamples.

Lesson 2-2  Conditional Statements 95

Concept Byte Logic and Truth Tables MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 2-2 Prepares for MG-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosve
tahbeoourtemlinseasbaonudt alinngelsesa.nAdlasongPlerse.pAalsroes for
activity PGr-eCpOa.Cre.1s0f,oGr-MCOA.FCS.1.9112.G-CO.3.10,
MMAP F2S.912.G-CO.3.11
MP 2

A compound statement combines two or more statements.

Key Concepts  Compound Statements

Compound How to Form It Example Symbols
Statement
s¿j
conjunction Connect two or more You will eat a sandwich You say “s and j.”
statements with and. and you will drink juice.
s¡j
disjunction Connect two or more You will eat a sandwich You say “s or j.”
statements with or. or you will drink juice.

A conjunction s ¿ j is true only when both s and j are true.
A disjunction s ¡ j is false only when both s and j are false.

1

For Exercises 1–4, use the statements below to construct the following
compound statements.

s: We will go to the beach.
j: We will go out to dinner.
t: We will go to the movies.

1. s ¿ j 2. s ¡ j 3. s ¡ ( j ¿ t) 4. (s ¡ j) ¿ t
9. x ¡ z
5. Write three of your own statements and label them s, j, and t. Repeat
Exercises 1–4 using your own statements.

For Exercises 6–9, use the statements below to determine the truth value of the
compound statement.

x: Emperor penguins are black and white.
y: Polar bears are a threatened species.
z: Penguins wear tuxedos.

6. x ¿ y 7. x ¡ y 8. x ¿ z

96 Concept Byte  Logic and Truth Tables

A truth table lists all the possible combinations of truth values for two or more statements.

Example p q pSq p¿q p¡q
T
p: Ohio is a state. T TTT
q: There are 50 states.

p: Georgia is a state. T F F F T
q: Miami is a state.

#p: 2 + 2 = 5 F T TFT

q: 2 2 = 4

p: 2 + 1 = 4 F F TFF
q: Dolphins are big fish.

2

To find the possible truth values of a complex statement such as (s ¿ j) ¡ ∼t, you can
make a truth table like the one below. You start with columns for the single statements
and add columns to the right. Each column builds toward the final statement. The
table below starts with columns for s, j, and t and builds to (s ¿ j) ¡ ∼t. Copy the table
and work with a partner to fill in the blanks.

s j t ϳt sٙ j (s ٙ j ) ٚ ϳt
T T T T 20. ?
T T F F T T
T F T 13. ? F 21. ?
T F F T
F T T F 17. ? 22. ?
F T F T F 23. ?
F F T 14. ? F F
10. ? 11. ? 12. ? 15. ? 24. ?
F 18. ?
16. ? 19. ?

25. Make a truth table for the statement (∼p ¡ q) ¿ ∼r.

26. a. Make a truth table for ∼(p ¿ q). Make another truth table for ∼p ¡ ∼q.
b. Make a truth table for ∼(p ¡ q). Make another for ∼p ¿ ∼q.
c. DeMorgan’s Law states that ∼(p ¿ q) = ∼p ¡ ∼q and that

∼(p ¡ q) = ∼p ¿ ∼q. How do the truth tables you made in parts (a) and (b)
show that DeMorgan’s Law is true?

Concept Byte  Logic and Truth Tables 97

2-3 Biconditionals and MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Definitions
Prepares for GM-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosvaebout lines
tahnedoarenmglsesa.bAolustolinPersepanadreasngfolers.GA-lCsOo.PCr.1e0p,aGre-CsOfo.Cr.11
MMAPF3S,.M91P2.6G, -MCOP.73.10, MAFS.912.G-CO.3.11
MP 3, MP 6, MP 7

Objective To write biconditionals and recognize good definitions

Look at the examples of insects and noninsects below. How would you
complete the following sentence: “If an animal is an insect, then . . .”?
Explain your reasoning.

Make sure you Insects Noninsects
consider all the
data. Ant Fly Beetle Spider Tick Centipede

MATHEMATICAL

PRACTICES

Lesson In the Solve It, you used conditional statements. A biconditional is a single true
statement that combines a true conditional and its true converse. You can write a
Vocabulary biconditional by joining the two parts of each conditional with the phrase if and only if.
• biconditional
Essential Understanding  A definition is good if it can be written as
a biconditional.

Problem 1 Writing a Biconditional

How else can What is the converse of the following true conditional? If the converse is
you write the also true, rewrite the statements as a biconditional.
biconditional? If the sum of the measures of two angles is 180, then the two angles
You can also write the
biconditional as “The are supplementary.
sum of the measures of
two angles is 180 if and Converse: If two angles are supplementary, then the sum of the measures
only if the two angles are of the two angles is 180.
supplementary.” The converse is true. You can form a true biconditional by joining the true
conditional and the true converse with the phrase if and only if.

Biconditional: Two angles are supplementary if and only if the sum of the
measures of the two angles is 180.

Got It? 1. What is the converse of the following true conditional? If the
converse is also true, rewrite the statements as a biconditional.

If two angles have equal measure, then the angles are c­ ongruent.

98 Chapter 2  Reasoning and Proof

Key Concept  Biconditional Statements
A biconditional combines p S q and q S p as p 4 q.

Example Symbols How to Read It
A point is a midpoint if and only if it divides a p4q “p if and only if q”
segment into two congruent segments.

You can write a biconditional as two conditionals that are converses.

Problem 2 Identifying the Conditionals in a Biconditional

How can you What are the two conditional statements that form this biconditional?

s eparate the A ray is an angle bisector if and only if it divides an angle into two
biconditional into
two parts? congruent angles.
Let p and q represent the following:
Identify the part before

and the part after the p: A ray is an angle bisector.
phrase if and only if.

q: A ray divides an angle into two congruent angles.

p S q: If a ray is an angle bisector, then it divides an angle into two congruent angles.

q S p: If a ray divides an angle into two congruent angles, then it is an angle bisector.

Got It? 2. What are the two conditionals that form this biconditional?
Two numbers are reciprocals if and only if their product is 1.

As you learned in Lesson 1-2, undefined terms such as point, line, and plane are the
building blocks of geometry. You understand the meanings of these terms intuitively.
Then you use them to define other terms such as segment.

A good definition is a statement that can help you identify or classify an object. A good
definition has several important components.

✔ A good definition uses clearly understood terms. These terms should be commonly
understood or already defined.

✔ A good definition is precise. Good definitions avoid words such as large, sort of,
and almost.

✔ A good definition is reversible. That means you can write a good definition as a true
biconditional.

Lesson 2-3  Biconditionals and Definitions 99

Problem 3 Writing a Definition as a Biconditional

How do you Is this definition of quadrilateral reversible? If yes, write it as a true biconditional.

determine whether Definition: A quadrilateral is a polygon with four sides.
raedveefrisnibitlieo?n is

Write the definition

as a conditional and Write a conditional. Conditional: If a figure is
the converse of the

conditional. If both are Write the converse. a quadrilateral, then it is a
true, the definition is polygon with four sides.
reversible.
Converse: If a figure is a

polygon with four sides, then

it is a quadrilateral.

The conditional and its converse Biconditional: A figure is a
are both true. The definition is quadrilateral if and only if it
reversible. Write the conditional is a polygon with four sides.
and its converse as a true
biconditional.

Got It? 3. Is this definition of straight angle reversible? If yes, write it
as a true biconditional.

A straight angle is an angle that measures 180.

One way to show that a statement is not a good definition is to find a counterexample.

Problem 4 Identifying Good Definitions
Multiple Choice  Which of the following is a good definition?

eHloimwincaanteyaonuswer A fish is an animal that swims. Giraffes are animals with very long necks.
choices? Rectangles have four corners. A penny is a coin worth one cent.
You can eliminate
an answer choice if Choice A is not reversible. A whale is a counterexample. A whale is an animal that
the definition fails to swims, but it is a mammal, not a fish. In Choice B, corners is not clearly defined. All
meet any one of the quadrilaterals have four corners. In Choice C, very long is not precise. Also, Choice C
components of a good is not reversible because ostriches also have long necks. Choice D is a good definition.
definition. It is reversible, and all of the terms in the definition are clearly defined and precise.
The answer is D.

Got It? 4. a. Is the following statement a good definition? Explain.
A square is a figure with four right angles.
b. Reasoning  How can you rewrite the statement “Obtuse angles have

greater measures than acute angles” so that it is a good definition?

100 Chapter 2  Reasoning and Proof

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. How can you write the following statement as two
4. Vocabulary  Explain how the term biconditional is
true conditionals? fitting for a statement composed of two conditionals.
Collinear points are points that lie on the
5. Error Analysis  Why is the following statement a
same line.
poor definition?
2. How can you combine the following statements as a Elephants are gigantic animals.
biconditional?
6. Compare and Contrast  Which of the following
If this month is June, then next month is July. statements is a better definition of a linear
If next month is July, then this month is June. pair? Explain.

3. Write the following definition as a biconditional. A linear pair is a pair of supplementary angles.
Vertical angles are two angles whose sides are
A linear pair is a pair of adjacent angles with
opposite rays. noncommon sides that are opposite rays.

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Each conditional statement below is true. Write its converse. If the converse is See Problem 1.
also true, combine the statements as a biconditional. See Problem 2.

7. If two segments have the same length, then they are congruent.

8. Algebra  If x = 12, then 2x - 5 = 19.

9. If a number is divisible by 20, then it is even.

10. Algebra  If x = 3, then 0 x 0 = 3.

11. In the United States, if it is July 4, then it is Independence Day.

12. If p S q is true, then ∼q S ∼p is true.

Write the two statements that form each biconditional.

13. A line bisects a segment if and only if the line intersects the segment only at
its midpoint.

14. An integer is divisible by 100 if and only if its last two digits are zeros.

15. You live in Washington, D.C., if and only if you live in the capital of the
United States.

16. A polygon is a triangle if and only if it has exactly three sides.

17. An angle is a right angle if and only if it measures 90.

18. Algebra  x2 = 144 if and only if x = 12 or x = -12.

Lesson 2-3  Biconditionals and Definitions 101

Test each statement below to see if it is reversible. If so, write it as a true See Problem 3.
biconditional. If not, write not reversible.

19. A perpendicular bisector of a segment is a line, segment, or ray that is
perpendicular to a segment at its midpoint.

20. Complementary angles are two angles with measures that have a sum of 90.

21. A Tarheel is a person who was born in North Carolina.

22. A rectangle is a four-sided figure with at least one right angle.

23. Two angles that form a linear pair are adjacent.

Is each statement below a good definition? If not, explain. See Problem 4.

24. A cat is an animal with whiskers. 25. The red wolf is an endangered animal.

26. A segment is part of a line. 27. A compass is a geometric tool.

28. Opposite rays are two rays 29. Perpendicular lines are two lines

that share the same endpoint. that intersect to form right angles.

B Apply 30. Think About a Plan  Is the following a good definition? Explain.

A ligament is a band of tough tissue connecting bones

or holding organs in place.

• Can you write the statement as two true conditionals?

• Are the two true conditionals converses of each other?

31. Reasoning  Is the following a good definition? Explain.
An obtuse angle is an angle with measure greater than 90.

32. Open-Ended  Choose a definition from a dictionary or from a glossary. Explain
what makes the statement a good definition.

33. Error Analysis  Your friend defines a right angle as an angle that is greater than an
acute angle. Use a biconditional to show that this is not a good definition.

34. Which conditional and its converse form a true biconditional?

If x 7 0, then 0 x 0 7 0. If x3 = 5, then x = 125.
If x = 19, then 2x - 3 = 35.
If x = 3, then x2 = 9.

Write each statement as a biconditional.
35. Points in Quadrant III have two negative coordinates.
36. When the sum of the digits of an integer is divisible by 9, the integer

is divisible by 9 and vice versa.
37. The whole numbers are the nonnegative integers.
38. A hexagon is a six-sided polygon.

102 Chapter 2  Reasoning and Proof

Language  For Exercises 39–42, use the chart below. Decide whether the
description of each letter is a good definition. If not, provide a counterexample
by giving another letter that could fit the definition.



39. The letter D is formed by pointing straight up with the finger beside the thumb and
folding the other fingers and the thumb so that they all touch.

40. The letter K is formed by making a V with the two fingers beside the thumb.

41. You have formed the letter I if and only if the smallest finger is sticking up and the
other fingers are folded into the palm of your hand with your thumb folded over
them and your hand is held still.

42. You form the letter B by holding all four fingers tightly together and pointing them
straight up while your thumb is folded into the palm of your hand.

Reading Math  Let statements p, q, r, and s be as follows:

p: ∠A and ∠B are a linear pair.
q: ∠A and ∠B are supplementary angles.
r: ∠A and ∠B are adjacent angles.
s: ∠A and ∠B are adjacent and supplementary angles.

Substitute for p, q, r, and s, and write each statement the way you would read it.

43. p S q 44. p S r 45. p S s 46. p 4 s

C Challenge 47. Writing  Use the figures to write a good definition of a line in spherical geometry.



Lines Not Lines
Lesson 2-3  Biconditionals and Definitions
103

48. Multiple Representations  You have illustrated true conditional statements with
Venn diagrams. You can do the same thing with true biconditionals. Consider the
following statement.

An integer is divisible by 10 if and only if its last digit is 0.

a. Write the two conditional statements that make up this biconditional.
b. Illustrate the first conditional from part (a) with a Venn diagram.
c. Illustrate the second conditional from part (a) with a Venn diagram.
d. Combine your two Venn diagrams from parts (b) and (c) to form a Venn diagram

representing the biconditional statement.
e. What must be true of the Venn diagram for any true biconditional statement?
f. Reasoning  How does your conclusion in part (e) help to explain why you can

write a good definition as a biconditional?

Standardized Test Prep

SAT/ACT 49. Which statement is a good definition?

Rectangles are usually longer than they are wide.
Squares are convex.
Circles have no corners.
Triangles are three-sided polygons.

50. What is the exact area of a circle with a diameter of 6 cm?

28.27 cm 9p m2 36p cm2 9p cm2

Extended 51. C onsider this true conditional statement.
Response If you want to buy milk, then you go to the store.

a. Write the converse and determine whether it is true or false.
b. If the converse is false, give a counterexample to show that it is false. If

the converse is true, combine the original statement and its converse as
a biconditional.

Mixed Review

Write the converse of each statement. See Lesson 2-2.
52. If you do not sleep enough, then your grades suffer.
53. If you are in the school chorus, then you have a good voice.
54. Reasoning  What is the truth value of the contrapositive of a true conditional?

G et Ready!  To prepare for Lesson 2-4, do Exercises 55–57.

What are the next two terms in each sequence? See Lesson 2-1.

55. 100, 90, 80, 70, c 56. 2500, 500, 100, 20, c 57. 1, 2, 0, 3, -1, c

104 Chapter 2  Reasoning and Proof

2 Mid-Chapter Quiz MathX

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FO

Do you know HOW? For Exercises 16–19, write the converse, inverse,
Use inductive reasoning to describe the pattern of each and contrapositive of each conditional statement.
sequence. Then find the next two terms. Determine the truth value of each statement. If it is
1. 1, 12, 123, 1234, c false, provide a counterexample.
2. 3, 4.5, 6.75, 10.125, c
3. 2, 3, 5, 7, 11, 13, c 16. If a figure is a circle with radius r, then its
circumference is 2pr.
Draw the next figure in each sequence.
4. 17. If an integer ends with 0, then it is divisible by 2.

5. 18. If you win the league championship game, then you
win the league trophy.
hsm11gmse_02mq_t00575
19. If a triangle has one right angle, then the other two
Find a counterexample for the conjecture. angles are complementary.
6. Three coplanar lines always make a triangle.
20. Write the two conditionals that make up this
7. hAlslmba1ll1sgamressep_he0r2ems. q_t00576 biconditional: An angle is an acute angle if and only
if its measure is between 0 and 90.
8. When it rains, it pours.
For Exercises 21–23, rewrite the definition as a
Identify the hypothesis and the conclusion of the biconditional.
conditional statements.
9. If the traffic light is red, then you must stop. 21. Points that lie on the same line are collinear.
10. If x 7 5, then x2 7 25.
11. If you leave your house, then you must lock the door. 22. Figures with three sides are triangles.

Rewrite the statements as conditional statements. 23. The moon is the largest satellite of Earth.
12. Roses are beautiful flowers.
13. Apples grow on trees. 24. Which of the following is a good definition?
14. Quadrilaterals have four sides. Grass is green.
15. The world’s largest trees are giant sequoias. Dinosaurs are extinct.
A pound weighs less than a kilogram.
A yard is a unit of measure exactly 3 ft long.

Do you UNDERSTAND?
25. Open-Ended  Describe a situation where you used a

pattern to reach a conjecture.

26. How does the word induce relate to the term
inductive reasoning?

27. Error Analysis  Why is the following not a good
definition? How could you improve it?

Rain is water.

Chapter 2  Mid-Chapter Quiz 105

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Prepares for GM-ACFOS.C.9.192  .PGro-CveOt.h3e.9o rePmrosvaebout lines
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MP 1, MP 2, MP 3, MP 4

Objective To use the Law of Detachment and the Law of Syllogism

Use each step You want to use the coupon to buy three different pairs of jeans. You
to write an have narrowed your choices to four pairs. The costs of the different
expression for the pairs are $24.99, $39.99, $40.99, and $50.00. If you spend as little
process as a whole. as possible, what is the average amount per pair of jeans that you will
pay? Explain.

MATHEMATICAL

PRACTICES

Lesson In the Solve It, you drew a conclusion based on several facts. You used deductive
reasoning. Deductive reasoning (sometimes called logical reasoning) is the process of
Vocabulary reasoning logically from given statements or facts to a conclusion.
• deductive
Essential Understanding  Given true statements, you can use deductive
reasoning reasoning to make a valid or true conclusion.
• Law of

Detachment
• Law of Syllogism

Property  Law of Detachment Symbols
If     p S q  is true
Law and   p    is true,
If the hypothesis of a true conditional is true, then
the conclusion is true. then       q   is true.

To use the Law of Detachment, identify the hypothesis of the given true conditional. If
the second given statement matches the hypothesis of the conditional, then you can
make a valid conclusion.

106 Chapter 2  Reasoning and Proof

Problem 1 Using the Law of Detachment

What can you conclude from the given true statements?

A Given:  If a student gets an A on a final exam, then the student will pass the course.
Felicia got an A on her history final exam.

  If a student gets an A on a final exam, then the student will pass the course.
Felicia got an A on her history final exam.

  The second statement matches the hypothesis of the given conditional. By the Law of
Detachment, you can make a conclusion.

  You conclude:  Felicia will pass her history course.

B Given:  If a ray divides an angle into two congruent angles, then the ray
iRsS a> nd iavnidgeles bjiAseRcBtosro. that jARS @ jSRB.
In part (C), the second
s tatement is not a   IRfSa> ray divides an angle into two congruent angles, then the ray is an angle bisector.
subset of the hypothesis.   divides ∠ARB so that ∠ARS ≅ ∠SRB.
Instead, it is a subset
of the conditional’s   The second statement matches the hypothesis of the given conditional. By the Law of
conclusion. Detachment, you can make a conclusion.

  You conclude:  RS> is an angle bisector.

C Given:  If two angles are adjacent, then they share a common vertex.
j1 and j2 share a common vertex.

  If two angles are adjacent, then they share a common vertex.
  ∠1 and ∠2 share a common vertex.

  The information in the second statement about ∠1 and ∠2 does not tell you if the
angles are adjacent. The second statement does not match the hypothesis of the
given conditional, so you cannot use the Law of Detachment. ∠1 and ∠2 could
be vertical angles, since vertical angles also share a common vertex. You cannot
make a conclusion.

Got It? 1. What can you conclude from the given information?
a. If there is lightning, then it is not safe to be out in the open.

Marla sees lightning from the soccer field.
b. If a figure is a square, then its sides have equal length.

Figure ABCD has sides of equal length.

Lesson 2-4  Deductive Reasoning 107

Another law of deductive reasoning is the Law of Syllogism. The Law of Syllogism
allows you to state a conclusion from two true conditional statements when the
conclusion of one statement is the hypothesis of the other statement.

Property  Law of Syllogism

Symbols Example
If   p S q   is true If it is July, then you are on summer vacation.
and    q S r  is true,
then   p S r  is true. If you are on summer vacation, then you work
at a smoothie shop.

You conclude: If it is July, then you work at a
smoothie shop.

When can you use the Problem 2 Using the Law of Syllogism
Law of Syllogism?
You can use the Law What can you conclude from the given information?

of Syllogism when A Given:  If a figure is a square, then the figure is a rectangle.
If a figure is a rectangle, then the figure has four sides.
the conclusion of
  I f a figure is a square, then the figure is a rectangle.
one statement is the If a figure is a rectangle, then the figure has four sides.

hypothesis of the other.   The conclusion of the first statement is the hypothesis of the second statement, so
you can use the Law of Syllogism to make a conclusion.

  You conclude: If a figure is a square, then the figure has four sides.

B Given:  If you do gymnastics, then you are flexible.
If you do ballet, then you are flexible.

  If you do gymnastics, then you are flexible.
  If you do ballet, then you are flexible.

  T he statements have the same conclusion. Neither conclusion is the hypothesis
of the other statement, so you cannot use the Law of Syllogism. You cannot
make a conclusion.

Got It? 2. What can you conclude from the given information? What is

your reasoning?

a. If a whole number ends in 0, then it is divisible by 10.

b. IIff aAwB>haonlde nAuDm> abreeroips pdoivsiistiebrlaeybsy, t1h0e,nththene it is divisible by 5. angle.
two rays form a straight

If two rays are opposite rays, then the two rays form a straight angle.

108 Chapter 2  Reasoning and Proof

You can use the Law of Syllogism and the Law of Detachment together to make
conclusions.

Problem 3 Using the Laws of Syllogism and Detachment

What can you conclude from the given information?

Given:  If you live in Accra, then you live in Ghana.

If you live in Ghana, then you live in Western Algeria Libya
Africa. Aissa lives in Accra. Sahara

If you live in Accra, then you live in Ghana. Mauritania Mali Niger
Burkina Nigeria
If you live in Ghana, then you live in Africa. Senegal Faso Chad
Gambia Sudan
Aissa lives in Accra. Guinea-Bissau Guinea

You can use the first two statements and the Sierra Leone Ivory Benin Central
Law of Syllogism to conclude: Liberia Coast Togo African Republic

Cameroon

Does the conclusion If you live in Accra, then you live in Africa. Equatorial Gabon Congo
make sense? Guinea Democratic
Accra is a city in Ghana, You can use this new conditional statement,
which is an African the fact that Aissa lives in Accra, and the Law of Republic of
nation. So if a person Congo
lives in Accra, then that
person lives in Africa. The Detachment to make a conclusion.
conclusion makes sense.
You conclude:  Aissa lives in Africa. Ghana Angola

Zambia

Got It? 3. a. What can you conclude from the given Accra Namibia
information? What is your reasoning? Botswana

If a river is more than 4000 mi long,

then it is longer than the Amazon. Lesotho

If a river is longer than the Amazon, South Africa

then it is the longest river in the world.

The Nile is 4132 mi long.

b. Reasoning  In Problem 3, does it matter whether you use the

Law of Syllogism or the Law of Detachment first? Explain.

Lesson Check Do You UNDERSTAND? MATHEMATICAL

Do You Know HOW? PRACTICES
If possible, make a conclusion from the given true
statements. What reasoning did you use? 4. Error Analysis  What is the error in the reasoning
below?
1. If it is Tuesday, then you will go bowling.
You go bowling. Birds that weigh more than 50 pounds
cannot fly. A kiwi cannot fly. So, a kiwi
2. If a figure is a three-sided polygon, then it is a triangle. weighs more than 50 pounds.
Figure ABC is a three-sided polygon.
5. Compare and Contrast  How is deductive reasoning
3. If it is Saturday, then you walk to work. different from inductive reasoning?
If you walk to work, then you wear sneakers.

hsm11gmse_0204_t00733

Lesson 2-4  Deductive Reasoning 109

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice If possible, use the Law of Detachment to make a conclusion. If it is not possible See Problem 1.
to make a conclusion, tell why. See Problem 2.
See Problem 3.
6. If a doctor suspects her patient has a broken bone, then she should take an X-ray.
Dr. Ngemba suspects Lilly has a broken arm.

7. If a rectangle has side lengths 3 cm and 4 cm, then it has area 12 cm2.
Rectangle ABCD has area 12 cm2.

8. If three points are on the same line, then they are collinear.
Points X, Y, and Z are on line m.

9. If an angle is obtuse, then it is not acute.
∠XYZ is not obtuse.

10. If a student wants to go to college, then the student must study hard.
Rashid wants to go to Pennsylvania State University.

If possible, use the Law of Syllogism to make a conclusion. If it is not possible to
make a conclusion, tell why.

STEM 11. Ecology  If an animal is a Florida panther, then its scientific name is Puma
concolor coryi.
If an animal is a Puma concolor coryi, then it is endangered.

12. If a whole number ends in 6, then it is divisible by 2.
If a whole number ends in 4, then it is divisible by 2.

13. If a line intersects a segment at its midpoint, then the line bisects the segment.
If a line bisects a segment, then it divides the segment into two
congruent segments.

14. If you improve your vocabulary, then you will improve your score on a
standardized test.
If you read often, then you will improve your vocabulary.

Use the Law of Detachment and the Law of Syllogism to make conclusions from
the following statements. If it is not possible to make a conclusion, tell why.

15. If a mountain is the highest in Alaska, then it is the highest in the United States.
If an Alaskan mountain is more than 20,300 ft high, then it is the highest in Alaska.
Alaska’s Mount McKinley is 20,320 ft high.

16. If you live in the Bronx, then you live in New York.
Tracy lives in the Bronx.
If you live in New York, then you live in the eleventh state to enter the Union.

17. If you are studying botany, then you are studying biology.
If you are studying biology, then you are studying a science.
Shanti is taking science this year.

110 Chapter 2  Reasoning and Proof