Geometry

1-2 Points, Lines, and CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Planes
GM-ACFOS.A.9.112  .KGn-oCwO.p1r.e1c isKendoewfinpirteiocnisseodfeafinngitleio,ncsircolfe,
apnergple,ncdiriculel,apr elirnpee,npdaircaullaerl line, apnadralilneel lsineeg,maenndtli.n.e.
sMegPm1e,nMt .P. 3. , MP 4, MP 6
MP 1, MP 3, MP 4, MP 6

Objective To understand basic terms and postulates of geometry

Make the figure at the right with
a pencil and a piece of paper. Is
the figure possible with a straight
arrow and a solid board? Explain.

Does how the
arrow goes
through the board
make sense?

MATHEMATICAL In this lesson, you will learn basic geometric facts to help you justify your answer to the
PRACTICES Solve It.

Essential Understanding  Geometry is a mathematical system built on accepted
facts, basic terms, and definitions.

Lesson In geometry, some words such as point, line, and plane are undefined. Undefined
terms are the basic ideas that you can use to build the definitions of all other figures
Vocabulary in geometry. Although you cannot define undefined terms, it is important to have a
• point general description of their meanings.
• line
• plane Key Concept  Undefined Terms
• collinear points
• coplanar Term Description How to Name It Diagram
• space
• segment A point indicates a location and You can represent a point by a dot A
• ray has no size. and name it by a capital letter,
• opposite rays such as A. ᐉ
• postulate
• axiom hsmB 11gmse_0
• intersection
A
A line is represented by a You can name a line by aansy<AtBw> o
straight path that extends in (proeiandt s“olinnethAeBl”in) eo,rs<BuAc>h, or by a
two opposite directions without
end and has no thickness. A line single lowercase letter, such as
contains infinitely many points.
line /.

A plane is represented by a flat You can name a plane by a capital A BP
surface that extends without end letter, such as plane P, or by at least
and has no thickness. A plane three points in the plane that do not all hsmC 11gmse_0102
contains infinitely many lines. lie on the same line, such as plane ABC.

Lesson 1-2  Points, Lines, and Planes 11

Points that lie on the same line are collinear points. Points and lines that lie in the same
plane are coplanar. All the points of a line are coplanar.

Why can figures have Problem 1 Naming Points, Lines, and Planes T
more than one name? A   What are two other ways to name <QT>? RQ S ᐉ
Lines and planes are PV
made up of many points. Two other ways to name <QT> are <TQ> and line m. N
You can choose any B   What are two other ways to name plane P?
two points on a line m
and any three or more Two other ways to name plane P are plane RQV and plane RSV.
noncollinear points in a
plane for the name. C   What are the names of three collinear points? What are the names of four
coplanar points?

P oints R, Q, and S are collinear. Points R, Q, S, and V are coplanar. hsm11gmse_0102_t00702.ai

Got It? 1. a. What are two other ways to name <RS>?
b. What are two more ways to name plane P?
c. What are the names of three other collinear points?
d. What are two points that are not coplanar with points R, S, and V?

The terms point, line, and plane are not defined because their definitions would
require terms that also need defining. You can, however, use undefined terms to define
other terms. A geometric figure is a set of points. Space is the set of all points in three
dimensions. Similarly, the definitions for segment and ray are based on points and lines.

Key Concept  Defined Terms

Definition How to Name It Diagram
A
A segment is part of a line You can name a segment by B
that consists of two endpoints its two endpoints, such as AB
and all points between them. (read “segment AB”) or BA.

A ray is part of a line that You can name a ray by its endpoint AB
consists of one endpoint and aans dABa>n(ortehaedr“proayinAt Bo”n).thTeheraoyr,dseurcohf
all the points of the line on points indicates the ray’s direction. hsm11gmse_0102_t00703.a
one side of the endpoint.

Opposite rays are two rays You can name opposite rays by hsmA 11gCmse_B0102_t00704.ai
that share the same endpoint their shared endpoint and any
and form a line. oasthCeAr>paonidntCoBn>.each ray, such

hsm11gmse_0102_t00705.ai

12 Chapter 1  Tools of Geometry

Problem 2 Naming Segments and Rays

How do you make A   What are the names of the segments in the figure at the right? DE F
sure you name all
the rays? The three segments are DE or ED, EF or FE, and DF or FD. hsm11gmse_0102_t00706.ai
Each point on the line is
an endpoint for a ray. At B   What are the names of the rays in the figure?
each point, follow the The four rays are DE> or DF>, ED>, EF>, and FD> or FE>.
line both left and right
to see if you can find a C   Which of the rays in part (B) are opposite rays?
second point to name The opposite rays are ED> and EF>.
the ray.

Got It? 2. Reasoning  EF> and FE> form a line. Are they opposite rays? Explain.

A postulate or axiom is an accepted statement of fact. Postulates, like undefined
terms, are basic building blocks of the logical system in geometry. You will use logical
reasoning to prove general concepts in this book.

You have used some of the following geometry postulates in algebra. For example, you
used Postulate 1-1 when you graphed equations such as y = 2x + 8. You graphed two
points and drew the line through the points.

Postulate 1-1 t
AB
Through any two points there is exactly one line.
Line t passes through points A and B. Line t is the only

line that passes through both points.

When you have two or more geometric figures, their intersectionhsm11g4myse_0102y_ϭt030x7Ϫ077.ai

is the set of points the figures have in common.

In algebra, one way to solve a system of two equations is to graph 2 (3, 2)
y ϭ Ϫ2x ϩ 8
them. The graphs of the two lines y = -2x + 8 and y = 3x - 7
intersect in a single point (3, 2). So the solution is (3, 2). This O1 3 5 7 x

illustrates Postulate 1-2. Ϫ2

Postulate 1-2

If two distinct lines intersect, then they intersect in exactly one point.hsm1A1gmsCe_01B02_t00708.ai
<AE> and <DB> intersect in point C.
DE

Lesson 1-2  Points, Lines, and Planes hsm11gmse_0102_t0103709.ai

There is a similar postulate about the intersection of planes. R W
ST
Postulate 1-3
If two distinct planes intersect, then they intersect in
exactly one line.

Plane RST and plane WST intersect in <ST>.

When you know two points that two planes have in common, Postulates 1‑1 and 1‑3 tell
you that the line through those points is the intersection of the planes.

hsm11gmse_0102_t00710.ai

Problem 3 Finding the Intersection of Two Planes D C
Each surface of the box at the right represents part of a plane. A B
What is the intersection of plane ADC and plane BFG?
H G
Plane ADC and plane BFG
EF

The intersection of the Find the points that the planes hsm11gmse_0102_t00711.ai
two planes have in common.

Is the intersection a D C Focus on plane ADC and plane BFG to
segment? A B see where they intersect.
No. The intersection of
the sides of the box is H G
a segment, but planes
continue without end. EF
The intersection is a line.
DC

hAsmH11gBmsGe_0Yp1ooui0nct2aB_ntas0ned0e p7tho1aint2tb.Cao.tih planes contain

EF
The planes intersect in <BC>.

Ghosmt It1?1g3m. bas..e WR_e0haa1st0oan2rie_nttg0h e0W7nha1ym3d.eaosiyoof utwoonplylanneeesdtthoaftiinndtetrwsoecctoimn <mBFo>?n points

to name the intersection of two distinct planes?

14 Chapter 1  Tools of Geometry

When you name a plane from a figure like the box in Problem 3, list the corner points in
consecutive order. For example, plane ADCB and plane ABCD are also names for the
plane on the top of the box. Plane ACBD is not.

Photographers use three-legged tripods to make sure that a
camera is steady. The feet of the tripod all touch the floor at the
same time. You can think of the feet as points and the floor as a
plane. As long as the feet do not all lie in one line, they will lie in
exactly one plane.

This illustrates Postulate 1-4.

Postulate 1-4 RS
PQ
Through any three noncollinear points there is exactly
one plane.

Points Q, R, and S are noncollinear. Plane P is the
only plane that contains them.

hsm11gmse_0102_t00714.ai

Problem 4 Using Postulate 1-4

How can you find the Use the figure at the right. M L
plane? JR
Try to draw all the lines A   What plane contains points N, P, and Q? Shade the plane. KQ
that contain two of the N P
three given points. You M L The plane on the bottom of the figure
will begin to see a plane JR contains points N, P, and Q.
form. KQ
N P

B   What plane contains points J, M, and Q? Shade the plane. hsm11gmse_0102_t00715.ai

ML

hJ sm11gmsKe_Q010Tt2hh_eetfip0glau0nr7ee1ctho6an.ttaapiianssspesoianttsaJs,lMan,tatnhdroQug. h

NP

Got It? 4. a. What plane contains points L, M, and N? Copy the figure in Problem 4

and shade the plane. name of a line that is coplanar with <JK> and <KL>?

hsm11gb.m Rseea_so0n1i0n2g_  Wt0h0a7t1is7t.haei

Lesson 1-2  Points, Lines, and Planes 15

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Use the figure at the right.
4. Vocabulary  A segment has endpoints R and S.
1. Wforh<aXtYa>?re two other names
X RS Y What are two names for the segment?
2. What are the opposite rays? 5. Are AB> and BA> the same ray? Explain.

3. What is the intersection of 6. dRreaawsoinnginogr nWamhyindgoayloinueussuecthwaosa<ErrFo>?wheads when
the two planes?
7. Compare and Contrast  How is naming a ray similar
to naming a line? How is it different?

Practice and Problehmsm-1S1oglmvsien_g01E0x2_etr0c0i7s1e8s.ai MATHEMATICAL

A Practice Use the figure at the right for Exercises 8–11. PRACTICES See Problem 1.
8. What are two other ways to name <EF>?
m
9. What are two other ways to name plane C? n EB F
CG
10. Name three collinear points.

11. Name four coplanar points.

Use the figure at the right for Exercises 12–14. See Problem 2.

12. Name the segments in the figure. hsmR11Sgmse_0102_t00719.ai

13. Name the rays in the figure. TW

14. a. Name the pair of opposite rays with endpoint T.
b. Name another pair of opposite rays.

Use the figure at the right for Exercises 15–26.

Name the intersection of each pair of planes. hsm11gmse_0102_t0S0e7e2P0r.aoiblem 3.

15. planes QRS and RSW 16. planes UXV and WVS XW
UV
17. planes XWV and UVR 18. planes TXW and TQU

Name two planes that intersect in the given line. T S
Q
19. <QU> 20. <TS> 21. <XT> 22. <VW> R
See Problem 4.
Copy the figure. Shade the plane that contains the given points.

23. R, V, W 24. U, V, W 25. U, X, S 26. T, U, V

hsm11gmse_0102_t00721.ai

16 Chapter 1  Tools of Geometry

B Apply Postulate 1-4 states that any three noncollinear points lie in exactly one plane.
Find the plane that contains the first three points listed. Then determine
whether the fourth point is in that plane. Write coplanar or noncoplanar Z YU
to describe the points. S
VX
27. Z, S, Y, C 28. S, U, V, Y
C
29. X, Y, Z, U 30. X, S, V, U

31. X, Z, S, V 32. S, V, C, Y

If possible, draw a figure to fit each description. Otherwise, write not possible.

33. four points that are collinear 34. two points that are noncollinear
35. three points that are noncollinear
36. three points that are nohnscmop1l1agnamr se_0102_t00722.a

37. aOnpden<E-FE>n, wdeitdh oDnreawofathfiegpuoreinwtsitohnpaolilntthsrBee, Cli,nDe,s.E, F, and G that shows <CD>, <BG>,

38. Think About a Plan  Your friend drew the diagram at the right to

prove to you that two planes can intersect in exactly one point.

Describe your friend’s error.
• How do you describe a plane?
• What does it mean for two planes to intersect each other?
• Can you define an endpoint of a plane?

39. Reasoning  If one ray contains another ray, are they the same ray?
Explain.

For Exercises 40–45, determine whether each statement is always, sometimes,
hsm11gmse_0102_t00723.ai
or never true.

40. <TQ> and <QT> are the same line.
41. JK> and JL> are the same ray.

42. Intersecting lines are coplanar.

43. Four points are coplanar.

44. A plane containing two points of a line contains the entire line.

45. Two distinct lines intersect in more than one point.

46. Use the diagram at the right. How many planes contain each line P Q

aacn.. d<<EFGpF>>oaainnntdd? ppooiinntt GP db.. <<PEPH>> and point E H G
and point G E F

e. Reasoning  What do you think is true of a line and a point not on

the line? Explain. (Hint: Use two of the postulates you learned in

this lesson.)

hsm11gmse_0102_t00724.ai

Lesson 1-2  Points, Lines, and Planes 17

In Exercises 47–49, sketch a figure for the given information. Then state the
postulate that your figure illustrates.

47. <AB> and <EF> intersect in point C.

48. The noncollinear points A, B, and C are all contained in plane N.
49. Planes LNP and MVK intersect in <NM>.

STEM 50. Telecommunications  A cell phone tower at A
point A receives a cell phone signal from the SE
southeast. A cell phone tower at point B receives
a signal from the same cell phone from due
west. Trace the diagram at the right and find
the location of the cell phone. Describe how
Postulates 1-1 and 1-2 help you locate the phone.

51. Estimation  You can represent the hands on a B
clock at 6:00 as opposite rays. Estimate the other W
11 times on a clock that you can represent as
opposite rays. HSM11GMSE_0102_a00319
2nd pass 01-02-09
52. Open-Ended  What are some basic words in Durke
English that are difficult to define?

Coordinate Geometry  Graph the points and state whether they are collinear.

53. (1, 1), (4, 4), ( -3, -3) 54. (2, 4), (4, 6), (0, 2) 55. (0, 0), ( -5, 1), (6, -2)

56. (0, 0), (8, 10), (4, 6) 57. (0, 0), (0, 3), (0, -10) 58. ( -2, -6), (1, -2), (4, 1)

C Challenge 59. How many planes contain the same three collinear points? Explain.

60. How many planes contain a given line? Explain.

61. a. Writing  Suppose two points are in plane P. Explain why the line containing the

points is also in plane P.

b. Reasoning  Suppose two lines intersect. How many planes do you think

contain both lines? Use the diagram at the right and your answer to part (a) AC
to explain your answer. B

Probability  Suppose you pick points at random from A, B, C, and D shown hsm11gmse_0102_t00725.ai
below. Find the probability that the number of points given meets the
condition stated.

62. 2 points, collinear D
63. 3 points, collinear A BC
64. 3 points, coplanar

hsm11gmse_0102_t00726.ai

18 Chapter 1  Tools of Geometry

Standardized Test Prep

SAT/ACT 65. Which geometric term is undefined? ray
segment plane
collinear

66. Which diagram is a net of the figure shown at the right?




hsm11gmse_0102_t00728.ai hsm11gmse_0102_t00727
hsm11gmse_0102_t00730.ai

67. You want to cut a block of cheese into four pieces. What is the least number

of cuts you need to make? 4h sm11gmse_0102_t 050731.ai A

2h sm11gmse_0102_t 03 0729.ai

Short 68. The figure at the right is called a tetrahedron.
Response a. Name all the planes that form the surfaces of the tetrahedron.
BD
b. Name all the lines that intersect at D.

C

Mixed Review

Make an orthographic drawing for each figure. Assume there are no hSseme 1L1esgsmonse1_-10. 102_t00
hidden cubes.

69. 70. 71.

Front Right Front Right Front Right

Simplify each ratio. See p. 891.

G 72e. t3hR0stemoa11d21 yg!m  sTeo_p0r1e0p2a_rte0f0o7r3L4e7.sa3sio. n31551xxh -3sm, d1o1gExmesrcei_se0s10725_–8t000. 73754..a ni 24h+nsnm11gmse_0102_t00736.ai

Simplify each absolute value expression. See p. 892.

75. 0 -6 0 76. 0 3.5 0 77. 0 7 - 10 0

Algebra  Solve each equation. 79. 3x + 9 + 5x = 81 See p. 894.
78. x + 2x - 6 = 6 80. w - 2 = -4 + 7w

Lesson 1-2  Points, Lines, and Planes 19

1-3 Measuring Segments CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MG-ACFOS.A.9.112  .KGn-oCwO.p1r.e1c isKendoewfinpirteiocnisseodfeafinngitleio,ncsircolfe,
apnergple,ncdiirculel,apr elirnpee,npdaircaullaerl line, apnadralilneel lsineeg,maenndtli.n.e.
sAelgsmoeGn-tG. P. .EA.Bls.6o MAFS.912.G-GPE.2.6

MP 2, MP 3, MP 4, MP 6

Objective To find and compare lengths of segments

On a freshwater fishing trip, you catch the fish below. By law, you
must release any fish between 15 and 19 in. long. You need to measure
your fish, but the front of the ruler on the boat is worn away. Can you
keep your fish? Explain how you found your answer.

Analyze the
problem to figure
out what you know
and what you need
to find next.

MATHEMATICAL In the Solve It, you measured the length of an object indirectly.

PRACTICES

Lesson Essential Understanding  You can use number operations to find and compare
the lengths of segments.
Vocabulary
• coordinate Postulate 1-5  Ruler Postulate AB
• distance
• congruent Every point on a line can be paired with a real ab
number. This makes a one-to-one correspondence coordinate of A coordinate of B
segments between the points on the line and the real numbers.
• midpoint The real number that corresponds to a point is called
• segment bisector the coordinate of the point.

The Ruler Postulate allows you to measure lengths of segments AB

using a given Cunonitsaidnedrt<oABfi>nadt distances between points on a hsma 11gmse_0103b_t00738.ai
number line. the right. The distance between
AB ϭ ͉ a Ϫ b ͉
points A and B is the absolute value of the difference of their

coordinates, or 0 a - b 0 . This value is also AB, or the length of AB.

hsm11gmse_0103_t00740.ai

20 Chapter 1  Tools of Geometry

Problem 1 Measuring Segment Lengths

What are you trying What is ST ? ST UV
to find?
ST represents the length The coordinate of S is -4. Ϫ6 Ϫ4 Ϫ2 0 2 4 6 8 10 12 14 16
of ST , so you are trying
to find the distance The coordinate of T is 8. Ruler Postulate
between points S and T.
ST = 0 - 4 - 8 0 Definition of distance

= 0 - 12 0 Subtract. hsm11gmse_0103_t00741.ai

= 12 Find the absolute value.

Got It? 1. What are UV and SV on the number line above?

Postulate 1-6  Segment Addition Postulate

If three points A, B, and C are collinear and B is AB BC C
between A and C, then AB + BC = AC.
AB
AC

Problem 2 Using the Segment Addition Poshtuslmat1e1gmse_0103_t00742.ai
Algebra  If EG = 59, what are EF and FG? 8x Ϫ 14 4x ϩ 1

EF G

EG = 59 EF and FG Use the Segment Addition Postulate to
EF = 8x - 14 write an equation.
FG = 4x + 1
hsm11gmse_0103_t00743.ai

EF + FG = EG Segment Addition Postulate
(8x - 14) + (4x + 1) = 59 Substitute.

12x - 13 = 59 Combine like terms.
12x = 72 Add 13 to each side.
x = 6 Divide each side by 12.

Use the value of x to find EF and FG.
EF = 8x - 14 = 8(6) - 14 = 48 - 14 = 34 Substitute 6 for x.
FG = 4x + 1 = 4(6) + 1 = 24 + 1 = 25

Got It? 2. In the diagram, JL = 120. What are JK and KL? 4x ϩ 6 7x ϩ 15

JK L

Lesson 1-3  Measuring Segmentsh sm11gmse_0103_t007442.a1i

When numerical expressions have the same value, you say that they are equal (=).
Similarly, if two segments have the same length, then the segments are
congruent (>) segments.

This means that if AB = CD, then AB ≅ CD. You can also say that if AB ≅ CD, then
AB = CD.

A 1 in. B A B

1 in. C D
CD

AB ϭ CD AB Х CD

As illustrated above, you can mark segments alike to show that they are congruent.
If there is more than one set of congruent segments, you can indicate each set with

the same numhbesrmo1f m1gamrkss.e_0103_t00745.ai

How do you know Problem 3 Comparing Segment Lengths
if segments are Are AC and BD congruent?
congruent?
Congruent segments A BC D E
have the same length. So
find and compare the Ϫ6 Ϫ4 Ϫ2 0 2 4 6 8 10 12 14 16
lengths of AC and BD.
AC = 0 -2 - 5 0 = 0 -7 0 = 7

Definition of distance

BD = 0 3 - 10 0 = 0 -7 0 = 7
Yes. AC = hBDsm, s1o1AgCm≅seB_D0.103_t00747.ai

Got It? 3. a. Use the diagram above. Is AB congruent to DE?

b. Reasoning  To find AC in Problem 3, suppose you subtract -2 from 5.
Do you get the same result? Why?

The midpoint of a segment is a point that divides the segment into two congruent
segments. A point, line, ray, or other segment that intersects a segment at its midpoint is
said to bisect the segment. That point, line, ray, or segment is called a segment bisector.

B is the midpoint ᐉ ᐉ is a segment
of AC. bisector of AC.

ABC

hsm11gmse_0103_t00749.ai

22 Chapter 1  Tools of Geometry

Problem 4 Using the Midpoint

How can you use Algebra  Q is the midpoint of PR. 6x Ϫ 7 5x ϩ 1
algebra to solve the What are PQ, QR, and PR?
problem? PQ R
The lengths of the
congruent segments Step 1 Find x.
are given as algebraic
expressions. You can set PQ = QR Definition of midpoint
the expressions equal to
each other. 6x - 7 = 5x + 1 Substitute. hsm11gmse_0103_t00762.ai

x - 7 = 1 Subtract 5x from each side.

x = 8 Add 7 to each side.

Step 2 Find PQ and QR.

PQ = 6x - 7 QR = 5x + 1

= 6(8) - 7 Substitute 8 for x. = 5(8) + 1

= 41 Simplify. = 41

Step 3 Find PR.

PR = PQ + QR Segment Addition Postulate

= 41 + 41 Substitute.

= 82 Simplify.

PQ and QR are both 41. PR is 82.

Got It? 4. a. Reasoning  Is it necessary to substitute 8 for x in the expression

for QR  in order to find QR? Explain.

b. U is the midpoint of TV . What are TU, UV, and TV? 8x ϩ 11 12x Ϫ 1

TUV

Lesson Check Do you UNDERSTAND? hsm11gmse_0103_t00766.ai
5. Vocabulary  Name two MATHEMATICAL
Do you know HOW? PRACTICES
Name each of the following. segment bisectors of PR.
ᐉ
ABCDE FG PQR S T
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
1. The point on DA> that is 2 units from D 6. Compare and Contrast  Describe 2 3 4 5 6
2. Two points that are 3 units from D the difference between saying that

3. Thhesmco1o1rdginmasteeo_f0t1h0e 3m_idt0p0o8in1t1o.faAi G two segments are congruent and saying that two

4. A segment congruent to AC segments have equal length. When would you use

each phrase?

7. Error Analysis  You and your friend hlivsem51m1igampasret._0103_t0081

He says that it is 5 mi from his house to your house
and -5 mi from your house to his house. What is
the error in his argument?

Lesson 1-3  Measuring Segments 23

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the length of each segment. See Problem 1.

8. AB 9. BD AB CD E

10. AD 11. CE Ϫ8 Ϫ6 13 7

Use the number line at the right for Exercises 12–14. See Problem 2.
See Problem 3.
12. If RS = 15 and ST = 9, then RT = ■. hR sm11gmse_010S3_t00820T.ai
13. If ST = 15 and RT = 40, then RS = ■.

14. Algebra  RS = 8y + 4, ST = 4y + 8, and RT = 15y - 9.
a. What is the value of y?

b. Find RS, ST, and RT. hsm11gmse_0103_t00821.ai

Use the number line below for Exercises 15–18. Tell whether the segments
are congruent.

LM NP Q

Ϫ10 Ϫ5 0 5 10 15

15. LN and MQ 16. MP and NQ 17. MN and PQ 18. LP and MQ

19. Aa.l gFheinbsdrmaX 1AA1.igsmthseem_i0d1p0o3in_tto0f0X7Y6.9.ai 3x 5x Ϫ 6 See Problem 4.

b. Find AY and XY. X AY

Algebra For Exercises 20–22, use the figure below. Find the value of PT.

20. PT = 5x + 3 and TQ = 7x - 9 hsm11gmse_0103_t00822

21. PT = 4x - 6 and TQ = 3x + 4 PTQ
22. PT = 7x - 24 and TQ = 6x - 2

B Apply On a number line, the coordinates of X, Y, Z, and W are −7, −3, 1, and 5,

respectively. Find the lengths of the two segments. Thhesmn t1e1llgwmhesteh_e0r 1th0e3y_t00824
are congruent.

23. XY and ZW 24. ZX and WY 25. YZ and XW

Suppose the coordinate of A is 0, AR = 5, and AT = 7. What are the possible
coordinates of the midpoint of the given segment?

26. AR 27. AT 28. RT

29. Suppose point E has a coordinate of 3 and EG = 5. What are the possible
coordinates of point G?

24 Chapter 1  Tools of Geometry

Visualization Without using your ruler, sketch a segment with the given length.
Use your ruler to see how well your sketch approximates the length provided.

30. 3 cm 31. 3 in. 32. 6 in. 33. 10 cm 34. 65 mm

35. Think About a Plan  The numbers labeled on the map of Florida are mile markers.
Assume that Route 10 between Quincy and Jacksonville is straight.

Quincy Monticello 95
199
Madison
181 10 251 Jacksonville
Tallahassee 225
283 303 Macclenny 10 357
Live Oak
Lake City 335

95

Suppose you drive at an average speed of 55 mi/h. How long will it take to get from

Live Oak to Jacksonville? HSM11GMSE_0103_a00326

• How dcaonayvoeuraugesespmeieledm, daisrtkaenrsceto, afninddtidmisetaanllcreeslabteetwtoeeeancphD2oonuindtrhktpesea?rs?s 01-05-09
• How

36. On a number line, A is at -2 and B is at 4. What is the coordinate of C, which is 2 of
3
the way from A to B?

Error Analysis  Use the highway sign for Exercises 37 and 38.

37. A driver reads the highway sign and says, “It’s 145 miles
from Mitchell to Watertown.” What error did the driver
make? Explain.

38. Your friend reads the highway sign and says, “It’s 71 miles
to Watertown.” Is your friend correct? Explain.

Algebra Use the diagram at the right for Exercises 39 and 40. A B

39. If AD = 12 and AC = 4y - 36, find the value of y. D C
Then find AC and DC. E

40. If ED = x + 4 and DB = 3x - 8, find ED, DB, and EB.

41. Writing  Suppose you know PQ and QR. Can you use the Segment Addition

Postulate to find PR? Explain. hsm11gmse_0103_t00825

C Challenge 42. C is the midpoint of AB, D is the midpoint of AC, E is the midpoint of AD,
F is the midpoint of ED, G is the midpoint of EF , and H is the midpoint

of DB. If DC = 16, what is GH?

43. a. Algebra  Use the diagram at the right. What 4x Ϫ 3

algebraic expression represents GK? 2x ϩ 3
b. If GK = 30, what are GH and JK? x

G HJ K

Lesson 1-3  Measuring Segments 25

hsm11gmse_0103_t00826

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 1

Look back at the information on page 3 about the riddle Cameron found in an

antique store. The page from the old riddle book is shown again below.

What sits in a corner but travels
around the world?

AG

B (7m + 3)° (8m − 18)° F
(11t − 17)° (5p + 2)°
H

(21s + 6)°
48°
C 5a + 12 D 9a − 12 E

Solve the riddle, today at the latest.
Arrange the variables from least to greatest.


a. What relationship between two segments can you state based on information in the
diagram? How does the diagram show this relationship?

b. Write and solve an equation to find the value of the variable a.
c. How can you be sure you solved the equation in part (b) correctly?

26 Chapter 1  Tools of Geometry

1-4 Measuring Angles CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MG-ACFOS.A.9.112  .KGn-oCwO.p1r.e1c isKendoewfinpirteiocnisseodfeafinngitleio,ncsircolfe,
apnergple,ncdiriculel,apr elirnpee,npdaircaullaerl line, apnadralilneel lsineeg,maenndtli.n.e.
sMegPm1e,nMt .P. 3. , MP 6
MP 1, MP 3, MP 6

Objective To find and compare the measures of angles

How can you Which angles below, if any, are the same size as the angle
use tools like a at the right? Describe two ways you can verify your answer.
protractor, ruler,
or tracing paper D
to help you solve BE
this?
G

A CF

MATHEMATICAL

PRACTICES
In this lesson, you will learn to describe and measure angles like the ones in the Solve It.

Lesson Essential Understanding  You can use number operations to find and compare
the measures of angles.
Vocabulary
• angle Key Concept  Angle
• sides of an angle
• vertex of an Definition How to Name It Diagram
An angle is formed by two rays
angle with the same endpoint. You can name an angle by B
• measure of an • its vertex, ∠A 1
The rays are the sides of the
angle angle. The endpoint is the • a point on each ray and the AC
• acute angle vertex of the angle. vertex, ∠BAC or ∠CAB aTrheeAsiBd>easnodf AthCe>.angle
• right angle The vertex is A.
• obtuse angle • a number, ∠1
• straight angle hsm11gmse_0104_t00
• congruent angles

When you name angles using three points, the vertex must go in the middle. exterior
interior
The interior of an angle is the region containing all of the points
between the two sides of the angle. The exterior of an angle is the region
containing all of the points outside of the angle.

Lesson 1-4  Measuring Angles 27

What rays form j1? Problem 1 Naming Angles JK
MJ> and MK> form ∠1. What are two other names for j1? 12 L
∠JMK and ∠KMJ are also names for ∠1.
M

Got It? 1. a. What are two other names for ∠KML?
b. Reasoning  Would it be correct to name any of the angles ∠M? Explain.

hsm11gmse_0104_t00830.ai

One way to measure the size of an angle is in degrees. To indicate the 62Њ

measure of an angle, write a lowercase m in front of the angle symbol. A
In the diagram, the measure of ∠A is 62. You write this as m∠A = 62.
In this book, you will work only with degree measures.

A circle has 360°, so 1 degree is 1 of a circle. A protractor forms half a
360
circle and measures angles from 0° to 180°.

Postulate 1-7  Protractor Postulate hsm11gmse_0104_t00831.ai

oCfoOnsBi>d. EerveOrBy>raanydofatphoeifnotrAmoOnAo>nceansibdee A 5013061020 70 80 90 100 110 6102050130
paired one to one with a real number 110 100 90 80 70
from 0 to 180.
1530014040 140401530016200

20
160

10 170
170 10

0 O 180
180 0
Binches
1 2 3 4 5 6

The Protractor Postulate allows you to find the measure of an angle. Consider the

dreiaagl nraummbbeelroswp.aTirheedmweitahsuOrCe> oafn∠d COODD>. Tishathteisa,hbifs moOlCu1>t1ecogvramrleusseepoo_fn0td1hse0w4dii_tfhfte0cr0,ean8nc3de2oO.afDith> e
corresponds with d, then m∠COD = 0 c - d 0 .

C 5013061020 70 80 90 100 110 6102050130
110 100 90 80 70

1530014040 c 140401530016200 D
d
20 6
160

10 170
170 10

0 O 180
180 0

inches 1 2 3 4 5

Notice that the Protractor Postulate and the calculation of an angle measure are
very similar to the Ruler Postulate and the calculation of a segment length.

hsm11gmse_0104_t00870.ai

28 Chapter 1  Tools of Geometry

You can classify angles according to their measures.

Key Concept  Types of Angles

acute angle right angle obtuse angle straight angle

xЊ xЊ xЊ xЊ
0 Ͻ x Ͻ 90 x ϭ 90 90 Ͻ x Ͻ 180 x ϭ 180

The symhbsoml  1 1i ngtmhesdei_a0gr1a0m4h_asbtm0o0v1e81i7ng1dm.iacsiaete_s0a1r0ig4hh_sttma0n01g81le7g.2sm.aie_0104_th0s0m87131.gasi me_0104_t00874.ai

Problem 2 Measuring and Classifying Angles

What are the measures of jLKN , jJKL, and jJKN ? Classify each angle
as acute, right, obtuse, or straight.

J

70 80 90 100 110 6102050130
110 100 90 80 70
L M5013061020
1530014040 140401530016200

20
160

10 170
170 10

0 180
180 0
H K Ninches
1 2 3 4 5 6

Do the classifications Use the definition of the measure of an angle to calculate each measure.
make sense?
Yes. In each case, the m∠LKN = 0 145 - 0 0 = 145; ∠LKN is obtuse.
classification agrees m∠JKL = 0 90hs-m141510g=m0 s-e5_500 1=0545_; t∠0J0K8L7i5s acute.
with what you see in the m∠JKN = 0 90 - 0 0 = 90; ∠JKN is right.
diagram.

Got It? 2. What are the measures of ∠LKH, ∠HKN , and ∠MKH? Classify
each angle as acute, right, obtuse, or straight.

Angles with the same measure are congruent angles. This means

that if m∠A = m∠B, then ∠A ≅ ∠B. You can also say that if A B
∠A ≅ ∠B, then m∠A = m∠B.
m∠A ϭ m∠B
You can mark angles with arcs to show that they are congruent. ∠A ഡ ∠B
If there is more than one set of congruent angles, each set is
marked with the same number of arcs.

hsm11gsme_0104_t00876.ai

Lesson 1-4  Measuring Angles 29

Look at the diagram. Problem 3 Using Congruent Angles D
What do the angle F
marks tell you? Sports  Synchronized swimmers form angles
with their bodies, as shown in the photo. E
The angle marks tell If mjGHJ = 90, what is mjKLM? B

you which angles are ∠GHJ ≅ ∠KLM because they
both have two arcs.
congruent. So, m∠GHJ = m∠KLM = 90.

C

A KM

Got It? 3. Use the photo in Problem 3. J L
If m∠ABC = 49, GH
what is m∠DEF ?

The Angle Addition Postulate is similar to the Segment Addition Postulate.

Postulate 1-8  Angle Addition Postulate

If point B is in the interior of ∠AOC, AB
then m∠AOB + m∠BOC = m∠AOC.

OC

Problem 4 Using the Angle Addition Postulate

How can you use the Algebra If mjRQT = 155, what are mjRQS anhdsmmj1T1QgSs?me_0104_t008S 77(3.axiϩ 14)Њ
expressions in the
diagram? m∠RQS + m∠TQS = m∠RQT Angle Addition Postulate (4x Ϫ 20)Њ T
The algebraic expressions
represent the measures 14x - 202 + 13x + 142 = 155 Substitute. RQ
of the smaller angles,
so they add up to the 7x - 6 = 155 Combine like terms.
measure of the larger
angle. 7x = 161 Add 6 to each side.

x = 23 Divide each side by 7. hsm11gmse_0104_t00878.ai

m∠RQS = 4x - 20 = 41232 - 20 = 92 - 20 = 72 Substitute 23 for x.
m∠TQS = 3x + 14 = 31232 + 14 = 69 + 14 = 83

Got It? 4. ∠DEF is a straight angle. What are m∠DEC C
and m∠CEF ?
(11x Ϫ 12)Њ (2x ϩ 10)Њ

DE F

30 Chapter 1  Tools of Geometry hsm11gmse_0104_t00879.ai

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Use the diagram for Exercises 1–3.
1. What are two other names for ∠1? C 4. Vocabulary  How many sides can two distinct,
2. Algebra If m∠ABD = 85, what is 1 xЊ D
A congruent angles share? Explain.
an expression to represent m∠ABC?
3. Classify ∠ABC. B 5. Error Analysis  Your classmate concludes from the
diagram below that ∠JKL ≅ ∠LKM. Is your

classmate correct? Explain.

hsm11gmse_0104_t00880.ai L
J KM

Practice and Problem-Solving Exercises PMRATAHCEMTAITCIhCEAsSLm11gmse_0104_t00881.ai

A Practice Name each shaded angle in three different ways. See Problem 1.

6. X 7. C 8. K L
YZ A 1 J 2

B M

Use the diagram below. Find the measure of each angle. Then classify the See Problem 2.

an9g. le∠haEssmAaFc1 u1tge,srmigeh_t,0o11b00t.u 4∠s_eD,t0oAr0Fs8 t8ra2i.gahit.hs1m1.1∠1BgAsEme_0104_t00883.ai hsmD11gsEme_0104_t00884.ai

12. ∠BAC 13. ∠CAE 14. ∠DAE 5013061020 70 80 90 100 110 6102050130
110 100 90 80 70

Draw a figure that fits each description. C 1530014040 140401530016200
15. an obtuse angle, ∠RST
20
160

10 170
170 10

16. an acute angle, ∠GHJ 0 180
17. a straight angle, ∠KLM 180 0
B A Finches
1 2 3 4 5 6

Use the diagram below. Complete each statement. See Problem 3.
18. ∠CBJ ≅ ■
19. ∠FJH ≅ ■ hsm11gsme_0104_t00885.ai
20. If m∠EFD = 75, then m∠JAB = ■.
21. If m∠GHF = 130, then m∠JBC = ■. C DE
F
G

B JH

A

Lesson 1-4  Mheassmur1in1ggAmngsele_s 0104_t11570.ai 31

22. If m∠ABD = 79, what are 23. ∠RQT is a straight angle. See Problem 4.

m∠ABC and m∠DBC? What are m∠RQS and m∠TQS?

S
DC

(5x ϩ 4)Њ (8x Ϫ 3)Њ (6x ϩ 20)Њ (2x ϩ 4)Њ

T QR

BA

B Apply Use a protractor. Measure and classify each angle.

24. 25. hsm11gsme2_60. 1 04_t00851.ai 27.
hsm11gsme_0104_t00850.ai

28. Think About a Plan  A pair of earrings has blue wedges that are all

the same size. One earring has a 25° yellow wedge. The other has

a••h 1sHH4m°ooy1wwe1lcdlgoaomwnthywseoeeua_dnu0ggse1le.e0Fam4ilng_edetab0tsrhu0aehr8teaos5snm2sogo.fl1aletv1hiemegtehseamaesrpureirrn_oeg0bosl1fera0mel4ba?l_tuete?0w0h8eds5mg3e.1.a1i gmse_0104_t205Њ0h8s5m4.1a1i gm14sЊe_0104_t00855.ai


Algebra  Use the diagram at the right for Exercises 29 and 30. AB C
Solve for x. Find the angle measures to check your work. O D

29. m∠AOB = 4x - 2, m∠BOC = 5x + 10, m∠COD = 2x + 14 Q
30. m∠AOB = 28, m∠BOC = 3x - 2, m∠AOD = 6x

31. If m∠MQV = 90, which expression can you use to find m∠VQP? P

V

m∠MQP - 90 m∠MQP + 90 hsm11Mgsme_0104_t00N856.ai

90 - m∠MQV 90 + m∠VQP

32. Literature  According to legend, King Arthur and his knights sat hsm11gsme_0104_t00857.ai
around the Round Table to discuss matters of the kingdom. The
photo shows a round table on display at Winchester Castle,
in England. From the center of the table, each section has the
same degree measure. If King Arthur occupied two of these
sections, what is the total degree measure of his section?

C Challenge Time  Find the angle measure of the hands of a clock at each time.

33. 6:00 34. 7:00 35. 11:00

36. 4:40 37. 5:20 38. 2:15

39. Open-Ended  Sketch a right angle with vertex V. Name it ∠1. Then sketch a 135°
angle that shares a side with ∠1. Name it ∠PVB. Is there more than one way to
sketch ∠PVB? If so, sketch all the different possibilities. (Hint: Two angles are the

same if you can rotate or flip one to match the other.)

32 Chapter 1  Tools of Geometry

40. Technology  Your classmate constructs an angle. Then he constructs a ray from the
vertex of the angle to a point in the interior of the angle. He measures all the angles
formed. Then he moves the interior ray as shown below. What postulate do the two
pictures support?



105Њ 105Њ
63Њ
81Њ
24Њ 42Њ

PERFORMANCE TASK

Apply Whhsamt1Y1gomus’ev_e01L0e4_atr0n08e5d8.ai hsm11gmse_0104_t00861PM.RaATiAHCEMTAITCICEASL

MP 6

Look back at the diagram on page 3 for the riddle Cameron found in an

antique store. Choose from the following words and equations to complete the

sentences below.

right congruent obtuse

21s + 6 = 48 21s + 6 + 48 = 90 21s + 6 + 48 = 180

s=6 s=2 s ≈ 1.7

a. In the riddle’s diagram, ∠HEF and ∠HED are ? angles.

b. The equation that correctly relates m∠HEF and m∠HED is ? .
c. The solution of the equation from part (b) is ? .

Lesson 1-4  Measuring Angles 33

1-5 Exploring Angle MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Pairs
Prepares for GM-ACFOS.A.9.112  .KGn-oCwO.p1r.e1c isKendoewfinpirteiocnisseof
danefginlei,ticoirncsleo,fpaenrpgelen,dciicruclea,rplienrep,epnadriaclulelal rlilninee., .p.arallel
lMinPe .1.,.MP 3, MP 4, MP 6
MP 1, MP 3, MP 4, MP 6

Objective To identify special angle pairs and use their relationships to find angle measures

It might help if The five game pieces at the right
you make a sketch form a square to fit back in the
of the pieces and box. Two of the shapes are already
cut them out. in place. Where do the remaining
pieces go? How do you know? Make
a sketch of the completed puzzle.

MATHEMATICAL

PRACTICES In this lesson, you will learn how to describe different kinds of angle pairs.

Essential Understanding  Special angle pairs can help you identify geometric
relationships. You can use these angle pairs to find angle measures.

Lesson Key Concept  Types of Angle Pairs

Vocabulary Definition Example
• adjacent angles
• vertical angles Adjacent angles are two ∠1 and ∠2, ∠3 and ∠4
• complementary coplanar angles with a common
side, a common vertex, and no 12 34
angles common interior points.
• supplementary

angles
• linear pair
• angle bisector

Vertical angles are two angles ∠1 and ∠2, ∠3 and ∠4 hsm11g1mse43_01025_t00888.ai
whose sides are opposite rays.

Complementary angles are two ∠1 and ∠2, ∠A and ∠B 1 47Њ B
angles whose measures have a ∠3 and ∠4, ∠B and ∠C
sum of 90. Each angle is called h2sm11Agmse4_30Њ 105_t00889.ai
the complement of the other.
3 4 137Њ
Supplementary angles are two
angles whose measures have a hsm11gmse_C0105_t00890.ai
sum of 180. Each angle is called
the supplement of the other.

34 Chapter 1  Tools of Geometry

Problem 1 Identifying Angle Pairs B

What should you look Use the diagram at the right. Is the statement true? Explain. A 28Њ C
f or in the diagram?
For part (A), check A jBFD and jCFD are adjacent angles. B 62Њ F
whether the angle pair No. They have a common side (FD>) F E
matches every part of C common 118Њ D
the definition of adjacent and a common vertex (F), but they D interior
angles. points
also have common interior points.
So ∠BFD and ∠CFD are not adjacent.

B jAFB and jEFD are vertical angles. hsm11gmse_0105_t00892.
NFEo>.aFnAd> aFnBd> aFrDe>naorte.
opposite rays, but B not opposite
So ∠AFB and ∠EFD A rays

are not vertical angles. hsm11gmF se_0105_t00893.ai

D
E

C jAFE and jBFC are complementary. B

Yes. m∠AFE + m∠BFC = 62 + 28 = 90. A 28Њ C
The sum of the angle measures is 90, so
hsm11gmF se_0105_t00894.ai
∠AFE and ∠BFC are complementary.
62Њ The sum of the
E measures is 90.

Got It? 1. Use the diagram in Problem 1. Is the statement true? Explain.
a. ∠AFE and ∠CFD are vertical angles.

bc.. ∠∠BBFFCD aanndd ∠∠DAFFBE aarree asudpjapcleenmteanhntsgamlreys1.. 1gmse_0105_t00895.ai

Concept Summary  Finding Information From a Diagram

There are some relationships you can assume to be true from a diagram that has
no marks or measures. There are other relationships you cannot assume directly.
For example, you can conclude the following from an unmarked diagram.
• Angles are adjacent.
• Angles are adjacent and supplementary.
• Angles are vertical angles.

You cannot conclude the following from an unmarked diagram.
• Angles or segments are congruent.
• An angle is a right angle.
• Angles are complementary.

Lesson 1-5  Exploring Angle Pairs 35

How can you get Problem 2 Making Conclusions From a Diagram 2

ia n dfoiarmgraatmio?n from What can you conclude from the information in the diagram? 1 3
54
L ook for relationships • ∠1 ≅ ∠2 by the markings.
be xeatwmepelen, angles. For • ∠3 and ∠5 are vertical angles.
look for • ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, ∠4 and ∠5, and ∠5 and ∠1

congruent angles and are adjacent angles.

a djacent angles. • ∠3 and ∠4, and ∠4 and ∠5 are adjacent supplementary angles.
So, m∠3 + m∠4 = 180 and m∠4 + m∠5 = 180 by the definition of
supplementary angles. hsm11gmseT_0105_t00896.ai

Got It? 2. Can you make each conclusion from the information in the P WQ
diagram? Explain. V

a. TW ≅ WV b. PW ≅ WQ
c. ∠TWQ is a right angle. d. TV bisects PQ.

A linear pair is a pair of adjacent angles whose noncommon sides Dhsm11gmse_0105_t00897.ai
are opposite rays. The angles of a linear pair form a straight angle.
AB C

Postulate 1-9  Linear Pair Postulate hsm11gmse_0105_t00898.ai
If two angles form a linear pair, then they are supplementary.

Problem 3 Finding Missing Angle Measures

Algebra  jKPL and jJPL are a linear pair, mjKPL = 2x + 24, and
mjJPL = 4x + 36. What are the measures of jKPL and jJPL?

∠KPL and ∠JPL are m∠KPL and Draw a diagram. Use the definition of supplementary
supplementary. m∠JPL angles to write and solve an equation.

Step 1 m∠KPL + m∠JPL = 180 Def. of supplementary angles L
(2x + 24) + (4x + 36) = 180 Substitute.
(2x ϩ 24)Њ (4x ϩ 36)Њ
6x + 60 = 180 Combine like terms. K PJ
6x = 120 Subtract 60 from each side.
x = 20 Divide each side by 6.

Step 2 Evaluate the original expressions for x = 20.
#m∠KPL = 2x + 24 = 2
#m∠JPL = 4x + 36 = 4 20 + 24 = 40 + 24 = 64 Substitute 20hfosrmx.11gmse_0105_t00899.ai
20 + 36 = 80 + 36 = 116

36 Chapter 1  Tools of Geometry

Got It? 3. a. Reasoning  How can you check your results in Problem 3?
b. ∠ADB and ∠BDC are a linear pair. m∠ADB = 3x + 14 and

m∠BDC = 5x - 2. What are m∠ADB and m∠BDC?

An angle bisector is a ray that divides an angle into two congruent X
Y
angles. Its endpoint is at the angle vertex. Within the ray, a segment
Z
with the same endpoint is also an aAnYg>lies bisector. The ray or segment A
bisects the angle. In the diagram, the angle bisector of ∠XAZ,

so ∠XAY ≅ ∠YAZ.

Problem 4 Using an Angle Bisector to Find Angle Measures
Multiple Choice  AC> bisects jDAB. If mjDAC = 58, what is mjDAB?

29 58 87 hsm 1116gmse_0105_t00900.ai

yD oruawvisaudaliiazgerwamhattoyhoeulp D 58Њ C Draw a diagram.

are given and what you AB

need to find.

m∠CAB = m∠DAC    Definition of angle bisector

= 58     Substitute.

m∠DhAsBm=1m1g∠mCAseB_+01m0∠5D_tA0C0 9 01 A.angi le Addition Postulate

= 58 + 58     Substitute.

= 116     Simplify.

The measure of ∠DAB is 116. The correct choice is D.

Got It? 4. KM> bisects ∠JKL. If m∠JKL = 72, what is m∠JKM?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES

Name a pair of the following types A E 5. Vocabulary  How does the term linear pair describe
of angle pairs. F how the angle pair looks?

1. vertical angles 6. Error Analysis  Your friend calculated 2xЊ
the value of x below. What is her error? 4xЊ
2. complementary angles

3. linear pair BC D 4x + 2x = 180
6x = 180
4. PB> bisects ∠RPT so that m∠RPB = x + 2 and x = 30
m∠TPB = 2x - 6. What is m∠RPT ?

hsm11gmse_0105_t00902.ai hsm11gmse_0105_

Lesson 1-5  Exploring Angle Pairs 37

hsm11gmse_0105_t00904.ai

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Use the diagram at the right. Is each statement true? Explain. See Problem 1.

7. ∠1 and ∠5 are adjacent angles. 4 51
8. ∠3 and ∠5 are vertical angles. 32
9. ∠3 and ∠4 are complementary.

10. ∠1 and ∠2 are supplementary.

Name an angle or angles in the diagram described by each of the following.

11. supplementary to ∠AOD hsm11Agmse_B0105_t00905.ai
12. adjacent and congruent to ∠AOE
13. supplementary to ∠EOA E
14. complementary to ∠EOD 60Њ O
15. a pair of vertical angles
DC

For Exercises 16–23, can you make each conclusion from the information in See Problem 2.

the diagram? Explain.

16. ∠J ≅ ∠D hsm11gE msFe_0105_t00906.ai
17. ∠JAC ≅ ∠DAC
18. m∠JCA = m∠DCA A

19. m∠JCA + m∠ACD = 180 J CD
20. AJ ≅ AD

21. C is the midpoint of JD.

22. ∠JAE and ∠EAF are adjacent and supplementary.

23. ∠EAF and ∠JAD are vertical angles. hsm11gmse_0105_t00907.ai

24. Name two pairs of angles that form a linear pair PQ See Problem 3.
in the diagram at the right.

25. ∠EFG and ∠GFH are a linear pair, L MN
m∠EFG = 2n + 21, and m∠GFH = 4n + 15.
What are m∠EFG and m∠GFH? FH See Problem 4.

26. Algebra  In the diagram, GH> bisects ∠FGI . hsm11gmse_0105_t00909.ai
a. Solve for x and find m∠FGH.
b. Find m∠HGI . (3x Ϫ 3)Њ (4x Ϫ 14)Њ
c. Find m∠FGI . G I

38 Chapter 1  Tools of Geometry hsm11gmse_0105_t00908.ai

B Apply Algebra  BD> bisects jABC. Solve for x and find mjABC.

27. m∠ABD = 5x, m∠DBC = 3x + 10

28. m∠ABC = 4x - 12, m∠ABD = 24

29. m∠ABD = 4x - 16, m∠CBD = 2x + 6

30. m∠ABD = 3x + 20, m∠CBD = 6x - 16

Algebra  Find the measure of each angle in the angle pair described.

31. Think About a Plan  The measure of one angle is twice the measure of
its supplement.

• How many angles are there? What is their relationship?
• How can you use algebra, such as using the variable x, to help you?

32. The measure of one angle is 20 less than the measure of its complement.

In the diagram at the right, mjACB = 65. Find each of the following. F A

33. m∠ACD 34. m∠BCD E CB
35. m∠ECD 36. m∠ACE

37. Algebra  ∠RQS and ∠TQS are a linear pair where m∠RQS = 2x + 4 D
and m∠TQS = 6x + 20.
hsm11gmse_0105_t00911.ai
a. Solve for x.
b. Find m∠RQS and m∠TQS.
c. Show how you can check your answer.

38. Writing  In the diagram at the right, are ∠1 and ∠2 adjacent?

Justify your reasoning. 12
39. Reasoning  When BX> bisects ∠ABC, ∠ABX ≅ ∠CBX . One student
1
claims there is always a related equation m∠ABX = 2 m∠ABC .

Another student claims the related equation is 2m∠ABX = m∠ABC.

Who is correct? Explain.

STEM 40. Optics  A beam of light and a mirror can be used hsm11gmse_0105_t00910.ai

to study the behavior of light. Light that strikes the Angle of Angle of
incidence reflection
mirror is reflected so that the angle of reflection
C
and the angle of incidence are congruent. In the
diagram, ∠ABC has a measure of 41.

a. Name the angle of reflection and find its AD

measure.

b. Find m∠ABD.

c. Find m∠ABE and m∠DBF . E BF
41. Reasoning  Describe all situations where vertical

angles are also supplementary.

Lesson 1-5  Exploring Angle Pairs 39

C Challenge Name all of the angle(s) in the diagram described by the following. N
MR
42. supplementary to ∠JQM 43. adjacent and congruent to ∠KMQ KL
J
44. a linear pair with ∠LMQ 45. complementary to ∠NMR

46. Coordinate Geometry  The x- and y-axes of the coordinate plane form QP
four right angles. The interior of each of the right angles is a quadrant of

the coordinate plane. What is the equation for the line that contains the

angle bisector of Quadrants I and III?

47. XXGC>> bbiisseeccttss ∠∠AEXXFB,, XD> bXiHse> cbtisse∠cAtsX∠CD, XXEB>.bIifsmec∠tsD∠XACX=D1, 6X,Ff>inbidsemct∠s G∠XEHXD. , hsm11gmse_0105_t00912.ai
and

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 3

Look back at the information on page 3 about the riddle Cameron found in an

antique store. The page from the old riddle book is shown again below.

What sits in a corner but travels
around the world?

AG

B (7m + 3)° (8m − 18)° F
(11t − 17)° (5p + 2)°
H

(21s + 6)°
48°
C 5a + 12 D 9a − 12 E

Solve the riddle, today at the latest.
Arrange the variables from least to greatest.

a. Name a pair of adjacent complementary angles in the diagram. Explain how you
know they are complementary.

b. Name a pair of nonadjacent complementary angles in the diagram.
c. In the Apply What You’ve Learned sections in Lessons 1-3 and 1-4, you found the

values of the variables a and s. Which variable’s value can you find next? Find the
value of this variable.

40 Chapter 1  Tools of Geometry

1 Mid-Chapter Quiz MathX

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

FO

Do you know HOW? 2. 16. a. Algebra  Find the value of x in the diagram below.
Draw a net for each figure.
b. Classify ∠ABC and ∠CBD as acute, right,
1. or obtuse.

C

2xЊ 3xЊ
ABD

Determine whether the given points are coplanar. Find the length of each segment.
If yes, name the plane. If no, explain.
17. PQ

3. Ah,sEm, F1,1agnmd Bse_01mAq_t0107h7s.mai11gmseB_01mq_t01145 1.a8i. RS hsmP 11gmQseR_01mqS_t01151T.ai
19. ST
4. D, C, E, and F EF Ϫ4 Ϫ2 0 2 4 6

5. H, G, F, and B D C 20. QT

6. A, E, B, and C HG Use the figure belohwsmfor1E1xgemrcsisee_s0211m–2q3._t01153.ai

7. Use the figure from Exercises 3–6. Name the F BE
intersection of each pair of planes.

a. plane AEFB and phlasnme C11BFgGmse_01mq_t01148.ai

b. plane EFGH and plane AEHD AC D

Use the figure below for Exercises 8–15. 21. Algebra  If AC = 4x + 5 and DC = 3x + 8,
find AD.
C
A RQ B 22. If m∠FCD = 1h30sman1d1mgm∠BsCe_D0=1m95q, f_int0d1m1∠55F.CaBi .

P 23. If m∠FCA = 50, find m∠FCE.

D Do you UNDERSTAND?
8. Give two other names for <AB>.
9. Give two other names for PR>. 24. Error Analysis  Suppose PQ = QR. Your friend says
that Q is always the midpoint of PR. Is he correct?
10. Give two othhesrmna1m1egsmfosre∠_C0P1Rm. q_t01150.ai Explain.

11. Name three collinear points. 25. Reasoning  Determine whether the following
12. Name two opposite rays. situation is possible. Explain your reasoning.
Include a sketch.

13. Name three segments. C <AoBl>liinnteearsrepcotsinptlsaCn,eFM, aantdCG. <AliBe> iannpdla<GnFe> M.
14. Name two angles that form a linear pair. do not intersect.
15. Name a pair of vertical angles.

Chapter 1  Mid-Chapter Quiz 41

Concept Byte Compass Designs MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 1-6 Prepares for MG-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl ake
fgoerommael tgriecocmoentsrtircucotinosntsruwctiitohnas vwairtihetay voafrtioetoylsoafnd
activity tmoeotlhsoadnsd(m . e. t.hdoydnsam( .i c. .gdeyonmaemtricicgseooftmweatreic. . .)
sMofPtw5are . . .)
MP 5

In Lesson 1-6, you will use a compass to construct geometric figures. You can
construct figures to show geometric relationships, to suggest new relationships,
or simply to make interesting geometric designs.

Step 1 Open your compass to about 2 in. Make a circle and mark the point at the
center of the circle. Keep the opening of your compass fixed. Place the
compass point on the circle. With the pencil end, make a small arc to
intersect the circle.

Step 2 Place the compass point on the circle at the arc. Mark another arc. Continue
around the circle this way to draw four more arcs—six in all.

Step 3 Place your compass point on an arc you marked on the circle. Place the
pencil end at the next arc. Draw a large arc that passes through the circle’s
center and continues to another point on the circle.

Step 4 Draw six large arcs in this manner, each centered at one of the six points
marked on the circle. You may choose to color your design.

Step 1 Step 2 Step 3 Step 4

Exercises

h1s. mIn1S1tgepm2s,ed_id0y1o0u6rasi_xt0h0m9ha7srm8k.oa1ni1tghme csierc_le01la0n6dap_rte0ci0s9el7y9oh.nasimthe1p1ogimntsweh_e0re10yo6ua_t0h0s9m801.1agi mse_0106a_t00981.ai

first placed your compass on the circle?
a. Survey the class to find out how many did.
b. Explain why your sixth mark may not have landed on your starting point.

2. Extend your design by using one of the six points on the circle as the center for
a new circle. Repeat Steps 1–4 with this circle. Repeat several times to make
interlocking circles

hsm11gmse_0106a_t00982.ai

42 Concept Byte  Compass Designs

1-6 Basic Constructions MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

GM-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl gaekoemfoertmricalcognesotmruecttriiocns with a
cvoanrisettryuoctfiotonoslswaitnhdamveatrhieotdyso(fctoomoplsaassndanmdestthroadigshtedge
(.c.o.)m. ApalssoaGn-dCsOtr.aAig.1htedge . . .). Also
MMAPF1S,.M91P2.3G, -MCOP.51,.1MP 7
MP 1, MP 3, MP 5, MP 7

Objective To make basic constructions using a straightedge and a compass

Draw jFGH. Fold your paper so that F
GH lies on top of GF. Unfold the H
paper. Label point J on the fold line G
in the interior of jFGH. How is GJ
Think about how related to jFGH? How do you know?
you might compare
angles without
measuring them.

MATHEMATICAL In this lesson, you will learn another way to construct figures like the one above.
PRACTICES

Lesson Essential Understanding  You can use special geometric tools to make a figure
that is congruent to an original figure without measuring. This method is more accurate
Vocabulary than sketching and drawing.
• straightedge
• compass A straightedge is a ruler with no markings on it. A compass is a geometric tool used to
• construction draw circles and parts of circles called arcs. A construction is a geometric figure drawn
• perpendicular using a straightedge and a compass.

lines
• perpendicular

bisector

Problem 1 Constructing Congruent Segments

Construct a segment congruent to a given segment.

Given:  AB AB
Construct:  CD so that CD ≅ AB

Why must the Step 1 Draw a ray with endpoint C. C
compass setting stay
the same? Step 2 Open the compass to the length of AB. hsm11gmse_0106_t009
Using the same compass
setting keeps segments Step 3 With the same compass setting, put the compass point AB
congruent. It guarantees on point C. Draw an arc that intersects the ray. Label
the point of intersection D. hsm11gmse_0106_t009
that the lengths of AB
CD ≅ AB CD
and CD are exactly the
same. hsm11gmse_0106_t009

Got It? 1. Use a straightedge to draw XY . Then construct RS so that RS = 2XY .

hsm11gmse_0106_t009

Lesson 1-6  Basic Constructions 43

Why do you need Problem 2 Constructing Congruent Angles A
points like B and C ?
B and C are reference Construct an angle congruent to a given angle. S
points on the original
angle. You can construct Given:  ∠A hsm11gmsBe_0106_t00922.ai
a congruent angle by
locating corresponding Construct:  ∠S so that ∠S ≅ ∠A A
points R and T on your
new angle. Step 1 hsm11gmsCe_0106_t00923.ai
Draw a ray with endpoint S.
Shsm11gmRse_0106_t00924.ai
Step 2
With the compass point on vertex A, draw an arc T
that intersects the sides of ∠A. Label the points of
intersection B and C. hS sm11gmRse_0106_t00925.ai

Step 3 T
With the same compass setting, put the compass
point on point S. Draw an arc and label its point of hS sm11gmsRe_0106_t00926.ai
intersection with the ray as R.
hsm11gmse_0106_t00927.ai
Step 4
Open the compass to the length BC. Keeping the B
same compass setting, put the compass point on R.
Draw an arc to locate point T.

DStreapw5ST>.

∠S ≅ ∠A

Got It? 2. a. Construct ∠F so that m∠F = 2m∠B.
b. Reasoning  How is constructing

a congruent angle similar to
constructing a congruent segment?

Perpendicular lines are two lines that intersect to form right angles.
thhesmrig1h1t,gmseA_0106_t0D0928.ai
T<AhBe> #sym<CbDo> lan#d m<CDea> n#s “<AisB>p. erpendicular to.” In the diagram at
CB

A perpendicular bisector of a segment is a line, segment, or ray that is

rpiegrhpte, n<EdF>icisutlahretpoetrhpeesnedgimcuelnart at its midpoint. In the diagram at the E midpoint
bisector of GH. The perpendicular of GH

bisector bisects the segment into two congruent segments. The

construction in Problem 3 will show you how this works. You will G hsm11FgmseH_0106_t00931
justify the steps for this construction in Chapter 4, as well as for the
other constructions in this lesson.

hsm11gmse_0106_t00932

44 Chapter 1  Tools of Geometry

Problem 3 Constructing the Perpendicular Bisector

Construct the perpendicular bisector of a segment.

Given:  AB A B
Construct:  <XY> so that <XY> is the perpendicular bisector of AB

Why must the Step 1 Ahsm11gmse_B0106_t00934.ai
compass opening be Put the compass point on point A and draw a long
greater than 12AB? arc as shown. Be sure the opening is greater than 21AB. X
If the opening is less than
12 AB, the two arcs will Step 2 hAsm11gmse_B0106_t00935
not intersect in Step 2. With the same compass setting, put the compass
point on point B and draw another long arc. Label Y
the points where the two arcs intersect as X and Y.

SDantredapw<X3Y<X>Ya>.sLMab, tehlethmeipdopionitnotfoifnAteBrs. ection of AB X
<XY> # AB at midpoint M, so <XY> is the perpendicular
bisector of AB. hsm11gmse_0106_t00936
AM B

Y

Got It? 3. Draw ST . Construct its perpendicular bisector.

Problem 4 Constructing the Angle Bisector hsm11gmse_0106_t00937

Construct the bisector of an angle. A

Given:  ∠A B
Construct:  AD>, the bisector of ∠A A

Step 1 hsm11gmC se_0106_t00938
Put the compass point on vertex A. Draw an arc
that intersects the sides of ∠A. Label the points
of intersection B and C.

Step 2 B
AD
Why must the arcs Put the compass point on point C and draw an arc. With the same
intersect? compass setting, draw an arc using point B. Be sure the arcs intersect. hsm11gmC se_0106_t00939
The arcs need to intersect Label the point where the two arcs intersect as D.
so that you have a point B
through which to draw SDtreapw3AD>. AD
a ray. AD> is the bisector of ∠CAB.
hsm11gmC se_0106_t00941
Got It? 4. Draw obtuse ∠XYZ. Then construct its bisector YP>.

Lesson 1-6  Basic Constructions hsm11gmse_01064_t500942

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
For Exercises 1 and 2, draw PQ. Use your drawing as
the original figure for each construction. 4. Vocabulary  What two tools do you use to make

constructions?

P Q 5. Compare and Contrast  Describe the difference in

accuracy between sketching a figure, drawing a figure

1. Construct a segment congruent to PQ. with a ruler and protractor, and constructing a figure.

2. Construct the perpendicular bisector of PQ. Explain.
6. Error Analysis  Your friend constructs <XY> so that it
3. Draw an obtuseh∠smJK1L1.gCmonssetr_u0c1t i0ts6b_its0e0ct9o4r.4.ai
is perpendicular to and contains the midpoint of AB.

H<XYe>.cWlaihmast that AB is the perpendicular bisector of
is his error?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice For Exercises 7–14, draw a diagram similar to the given one. Then do the
construction. Check your work with a ruler or a protractor.

7. Construct XY congruent to AB. A B See Problem 1.
8. Construct VW so that VW = 2AB.
See Problem 2.
9. Construct DE so that DE = TR + PS. T RP S See Problem 3.
10. Construct QJ so that QJ = TR - PS.
11. Construct ∠D so that ∠D ≅ ∠C. hsm11gmse_0106_t00945.ai

C

12. Construct ∠F so that m∠F = 2m∠C. hsm11gmse_0106_t00947.ai

13. Construct the perpendicular bisector of AB.

14. Construct the perpendicular bisector of TR.

15. Draw acute ∠PQR. Then construct its bisector. See Problem 4.

16. Draw obtuse ∠XQZ. Then construct its bisector. hsm11gmse_0106_t00948.ai

B Apply Sketch the figure described. Explain how to construct it. Then do the construction.
17. <XY> # <YZ>
18. ST> bisects right ∠PSQ.

19. Compare and Contrast  How is constructing an angle bisector similar to
constructing a perpendicular bisector?

46 Chapter 1  Tools of Geometry

20. Think About a Plan  Draw an ∠A. Construct an angle whose measure is 41m∠A.
• How is the angle you need to construct related to the angle bisector of ∠A?
• How can you use previous constructions to help you?

21. Answer the questions about a segment in a plane. Explain each answer.
a. How many midpoints does the segment have?
b. How many bisectors does it have?
c. How many lines in the plane are its perpendicular bisectors?
d. How many lines in space are its perpendicular bisectors?

For Exercises 22–24, copy j1 and j2. Construct each angle described.

22. ∠B; m∠B = m∠1 + m∠2

23. ∠C; m∠C = m∠1 - m∠2 12
24. ∠D; m∠D = 2m∠2

25. Writing  Explain how to do each construction with a compass and straightedge.
a. Draw a segment PQ. Construct the midpoint of PQ.

b. Divide PQ into four congruent segmhesnmts1. 1gmse_0106_t00950.ai

26. a. Draw a large triangle with three acute angles. Construct the bisectors of the three
angles. What appears to be true about the three angle bisectors?

b. Repeat the constructions with a triangle that has one obtuse angle.
c. Make a Conjecture  What appears to be true about the three angle bisectors of

any triangle?

Use a ruler to draw segments of 2 cm, 4 cm, and 5 cm. Then construct each triangle
with the given side measures, if possible. If it is not possible, explain why not.

27. 4 cm, 4 cm, and 5 cm 28. 2 cm, 5 cm, and 5 cm

29. 2 cm, 2 cm, and 5 cm 30. 2 cm, 2 cm, and 4 cm

31. a. Draw a segment, XY . Construct a triangle with sides congruent to XY .

b. Measure the angles of the triangle.
c. Writing  Describe how to construct a 60° angle using what you know. Then

describe how to construct a 30° angle.

32. Which steps best describe how to construct the pattern

at the right?

Use a straightedge to draw the segment and then a

compass to draw five half circles.

Use a straightedge to draw the segment and then a compass to draw six

half circles. to draw five half circles and then a hsm11gmse_0106_t00951.ai
Use a compass
straightedge to join their ends.

Use a compass to draw six half circles and then a straightedge to join their ends.

Lesson 1-6  Basic Constructions 47

C Challenge 33. Study the figures. Complete the definition of a line r t
perpendicular to a plane: A line is perpendicular to a M P
plane if it is ? to every line in the plane that ? .

34. a. Use your compass to draw a circle. Locate three

points A, B, and C on the circle.

b. Draw AB and BC. Then construct the perpendicular Line r Ќ plane M. Line t is not Ќ plane P.
bisectors of AB and BC.

c. Reasoning  Label the intersection of the two perpendicular bisectors as point O.

What do you think is true about point O?

35. Two triangles are congruent if each side and each angle of ohnsemtr1ia1ngglme issec_o0n1g0ru6e_ntt00952

to a side or angle of the other triangle. In Chapter 4, you will learn that if each side

of one triangle is congruent to a side of the other triangle, then you can conclude

that the triangles are congruent without finding the angles. Explain how you can

use congruent triangles to justify the angle bisector construction.

Standardized Test Prep hsm11gmse_0106_t00953

SAT/ACT 36. What must you do to construct the midpoint of a segment?

Measure half its length. Measure twice its length.

Construct an angle bisector. Construct a perpendicular bisector.

37. Given the diagram at the right, what is NOT a reasonable name for the angle? B A

∠ABC ∠CBA C
x
Short ∠B ∠ACB
Response
38. M is the midpoint of XY . Find the value of x. Show your work. x2 Ϫ 2

XM Y

Mixed Review hsm11gmse_0106_t00954.ai

39. ∠DEF is the supplement of ∠DEG with m∠DEG = 64. What is m∠DEFh?sm11gmse_0S1ee06L_est0so0n951-65..ai

40. m∠TUV = 100 and m∠VUW = 80. Are ∠TUV and ∠VUW a linear
pair? Explain.

Find the length of each segment. ABC See Lesson 1-3.
41. AC 42. AD D
43. CD 44. BC Ϫ6 Ϫ4 Ϫ2 0 24

G et Ready!  To prepare for Lesson 1-7, do Exercises 45–47.

Algebra  Evaluate each expression for a = 6 and b = −8. See p. 890.
46. 2a2 + b2 hsm11gmse_04170. 6a_+2t0b0957.ai
45. (a - b)2

48 Chapter 1  Tools of Geometry

Concept Byte Exploring CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Constructions
Use With Lesson 1-6 GM-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl gaekoemfoertmricalcognesotmruecttriiocns with a
cvoanrisettryuoctfiotonoslswaitnhdamveatrhieotdyso(fctoomoplsaassndanmdestthroadigshtedge
technology (.c.o.)m. pass and straightedge . . .).

MP 5

You can use Draw tools or Construct tools in geometry software to make points, lines, MATHEMATICAL
and planes. A figure made by Draw has no constraints. When you manipulate, or try
to change, a figure made by Draw, it moves or changes size freely. A figure made by PRACTICES
Construct is related to an existing object. When you manipulate the existing object,
the constructed object moves or resizes accordingly.

In this Activity, you will explore the difference between Draw and Construct.

DB H
DCoranwstrAuBctaGn,daCnoynpsotrinutcot nthEeFp.eDrpreanwd<HicGu>l.ar bisector <DC>. Then Draw EF and F
C
1. Find EG, GF, and m∠HGF . Try to drag G so that EG = GF . Try to drag H A EG
so that m∠HGF = 90. Were you able to draw the perpendicular bisector
of EF ? Explain.

2. Drag A and B. Observe AC, CB, and m∠DCB. Is <DC> always the
perpendicular bisector of AB no matter how you manipulate the figure?

3. DEFra agnEda<HndG>Fd.iOffebrseenrvt efrEomG, tGhFe,raenladtimon∠shHiGpFb.eHtwoewenisAthBearneldat<DioCn>?ship between

4. fWursoremityeotahuedr edreselcsarctiirpoitpnitosinhoniopftbotheetexwpgeelaeninenrAwaBlhdyainftfhdeer<DerneCcl>a.etiboentswheiepnbDetrwaweeannEdFCaonndst<HruGc>td. Tifhfheernssm11gmse_0106hbs_mt01019g8m3.saei _0106b_t0

Exercises NQ
OP
5. a. Draw ∠NOP. Draw OQ> in the interior of ∠NOP. Drag Q until
m∠NOQ = m∠QOP. J

b. MIs aOnQip> aullwataeytshtehefigaunrgeleanbdiseocbtsoerrovef ∠thNeOdPiff?erent angle measures. hsm11gmseM_0106b_t00986.ai

6. a. Draw ∠JKL. angle bisector, KM>. KL
b. Construct its

c. MIs aKnMip> uallwataeytshtehfeigaunrgeleanbdisoecbtsoerrvoef the different angle measures.
∠JKL?

d. How can you manipulate the figure on the screen so that it shows

a right angle? Justify your answer.

Concept Byte  Exploring Chosnmst1ru1cgtimonsse _0106b_t00987.ai 49

1-7 Midpoint and Distance MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
in the Coordinate Plane
Prepares for MG-AGFPSE..9B1.24. GU-sGePcEo.o2r.d4in  aUtsees to prove
csiomoprdleingaeteosmteotrpicrotvheeosirmemplse.g.e.ometric theorems . . .
Prepares for MG-AGFPSE..9B1.27. GU-sGePcEo.o2r.d7in  aUtsees to
coomrpduintae tpeesrtiomceotemrspu. t.e. apnedrimareetaesrs. . . .Aalnsdo aGre-GasP.E..B. .6
AMlPso1M, MAPFS3.,9M12P.G4-GPE.2.6
MP 1, MP 3, MP 4

Objectives To find the midpoint of a segment
To find the distance between two points in the coordinate plane

Try drawing the In a video game, two ancient structures shoot LEFT RIGHT
situation on graph light beams toward each other to form a UP DOWN
paper if you are time portal. The portal forms exactly
having trouble halfway between the two structures. Your
visualizing it. character is on the grid shown as a blue dot.
How do you direct your character to the
portal? Explain how you found your answer.

MATHEMATICAL In this lesson, you will learn how to find midpoints and distance on a grid like the one

PRACTICES in the Solve It.

Essential Understanding  You can use formulas to find the midpoint and length
of any segment in the coordinate plane.

Key Concept  Midpoint Formulas

Description Formula Diagram

On a Number Line The coordinate of the + b. AMB
a 2 a aϩb b
The coordinate of the midpoint M of AB is
midpoint is the average or 2
mean of the coordinates of
the endpoints.

In the Coordinate Plane Given AB where A(x1, y1) hsm1y21gmse_0107_Bt00958.ai
and B(x2, y2), the coordinates y1 ϩ y2 M
The coordinates of the
midpoint are the average of the midpoint of AB are 2 A
of the x‑coordinates
and the average of the ( )Mx1+x2, y1 + y2 y1
y‑coordinates of the 2 2
endpoints. . O x1 x1 ϩ x2 x2

2

50 Chapter 1  Tools of Geometry

Problem 1 Finding the Midpoint

Which Midpoint A AB has endpoints at −4 and 9. What is the AB
Formula do you use? coordinate of its midpoint?
If the endpoints are real Ϫ8 Ϫ6 Ϫ4 Ϫ2 0 2 4 6 8 10 12
numbers, use the formula Let a = - 4 and b = 9.
for the number line. If
they are ordered pairs, M = a + b = -4 + 9 = 5 = 2.5
use the formula for the 2 2 2
coordinate plane.
The coordinate of the midpoint of AB is 2.5. hsm11gmse_0107_t00960.ai

B EF has endpoints E(7, 5) and F(2, −4). What are the y E(7, 5)
coordinates of its midpoint M? 4
2 x
Let E(7, 5) be (x1, y1) and F(2, - 4) be (x2, y2). 68
O2
x-coordinate of M = x1 + x2 = 7 + 2 = 9 = 4.5 Ϫ2
2 2 2

y-coordinate of M = y1 + y2 = 5 + ( - 4) = 1 = 0.5
2 2 2

The coordinates of the midpoint of EF are M(4.5, 0.5). Ϫ4 F(2, ؊4)

Got It? 1. a. JK has endpoints at -12 and 4 on a number
line. What is the coordinate of its midpoint?

b. What is the midpoint of RS with endpoints R(5, -10) and S(3, 6)?

hsm11gmse_0107_t00961.ai

When you know the midpoint and an endpoint of a segment, you can use the
Midpoint Formula to find the other endpoint.

Problem 2 Finding an Endpoint

How can you find the The midpoint of CD is M(−2, 1). One endpoint is C(−5, 7). C y
coordinates of D? What are the coordinates of the other endpoint D?
Use the Midpoint 6
Formula to set up an
equation. Split that Let M( -2, 1) be (x, y) and C( -5, 7) be (x1, y1). Let the coordinates 4
equation into two of D be (x2, y2).
equations: one for the
x‑coordinate and one for ( -2, 1) = a -5 + x2, 7 + y2 b 2
the y‑coordinate. 2 2 Mx

x y Ϫ6 Ϫ4 Ϫ2 O 2

- 2 = -5 + x2 Use the Midpoint Formula. 1 = 7 + y2
2 2
Ϫ4
- 4 = - 5 + x2 Multiply each side by 2. 2 = 7 + y2

1 = x2 Simplify. - 5 = y2

The coordinates of D are (1, -5).

hsm11gmse_0107_t00962.ai

Got It? 2. The midpoint of AB has coordinates (4, -9). Endpoint A has coordinates
( -3, -5). What are the coordinates of B?

Lesson 1-7  Midpoint and Distance in the Coordinate Plane 51

In Lesson 1-3, you learned how to find the distance between two points on a number
line. To find the distance between two points in a coordinate plane, you can use the
Distance Formula.

Key Concept  Distance Formula

The distance between two points A(x1, y1) and B(x2, y2) is y d B
d = 2(x2 - x1)2 + (y2 - y1)2. x2 Ϫ x1 y2 Ϫ y1
y2
x
y1 A
x2
O x1

TwhheicDhi sytoaun cweilFl ostrumduyllaaitserbiansethdios nbothoek.PWythheangoyroeuanusTehtehoeremhs, m11gmse_0107c_t009b64.ai

Distance Formula, you are really finding the length of a side of a a

right triangle. You will verify the Distance Formula in Chapter 8. a2 ϩ b2 ϭ c2

Problem 3 Finding Distance

What is the distance between U(−7, 5) and V(4, −3)? Round to the neahrsemst1te1ngthm. se_0107_t00965.ai

Let U( - 7, 5) be (x1, y1) and V(4, - 3) be (x2, y2).

d = 2(x2 - x1)2 + (y2 - y1)2 Use the Distance Formula.

What part of a right = 2(4 - ( - 7))2 + ( - 3 - 5)2 Substitute.
triangle is UV ?
UV is the hypotenuse of = 2(11)2 + ( - 8)2 Simplify within the parentheses. 13.6
a right triangle with legs
of length 11 and 8. ؊

U = 1121 + 64 Simplify. 000000
8 = 1185 111111
222222
11 V 185 13.60147051 Use a calculator. 333333
444444
To the nearest tenth, UV = 13.6. 555555
666666
777777
888888
999999

Got It? 3. a. SR has endpoints S(-2, 14) and R(3, -1). What is SR to the hsm11gmse_0107_t00966.ai
hsm11gmse_0107_t00967.ai b. Rneeaarseosntitnegn thIn? Problem 3, suppose you let V(4, -3) be (x1, y1)

and U( -7, 5) be (x2, y2). Do you get the same result? Why?

52 Chapter 1  Tools of Geometry

Problem 4 Finding Distance

Recreation  On a zip-line course, you are harnessed to a cable that travels through
the treetops. You start at Platform A and zip to each of the other platforms. How far
doPyrooubtlreamvel 4from Platform B to Platform C? Each grid unit represents 5 m.

A 20 y D F
C 10

x
50 30 10 O 10 20 30 40 50

10

B 20 E

Where’s the right Let Platform B( - 30, - 20) be (x1, y1) and Platform C( - 15, 10) be (x2, y2).
triangle?

The lengths of the legs

of the right triangle are d = 2(x2 - x1)2 + (y2 - y1)2 Use the Distance Formula.
15 and 30. There are two
possibilities: = 2( - 15 - ( - 30))2 + (10 - ( - 20))2 Substitute.

15 C C

30 or 30 = 2152 + 302 = 1225 + 900 = 11125 Simplify.

B B 15 1125 33.54101966 Use a calculator.

You travel about 33.5 m from Platform B to Platform C.

Got It? 4. How far do you travel from Platform D to Platform E?

hsm11gmse_0107_t01710.ai

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. RS has endpoints R(2, 4) and S( -1, 7). What are
4. Reasoning  How does the Distance Formula ensure that the
the coordinates of its midpoint M?
distance between two different points is positive?
2. The midpoint of BC is (5, -2). One endpoint is
B(3, 4). What are the coordinates of endpoint C? 5. Error Analysis  Your d = V(1 - 8)2 + (5 - 3)2
friend calculates the = V(-7)2 + 22
3. What is the distance between points K( -9, 8) distance between points = V49 + 4
and L( -6, 0)? Q(1, 5) and R(3, 8). What = V53 ≈ 7.3
is his error?

Lesson 1-7  Midpoint and Distance in the Coordinate Plane 53

hsm11gmse_0107_t00968.ai

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the coordinate of the midpoint of the segment with the given endpoints. See Problem 1.
9. -8 and -12
6. 2 and 4 7. -9 and 6 8. 2 and -5

Find the coordinates of the midpoint of HX .

10. H(0, 0), X(8, 4) 11. H( -1, 3), X(7, -1) 12. H(13, 8), X( -6, -6)

13. H(7, 10), X(5, -8) 14. H( -6.3, 5.2), X(1.8, -1) ( ) ( )15. H5 21,-43,X 2 14, - 1 1
4 4

The coordinates of point T are given. The midpoint of ST is (5, −8). Find the See Problem 2.
coordinates of point S.

16. T(0, 4) 17. T(5, -15) 18. T(10, 18)

19. T( -2, 8) 20. T(1, 12) 21. T(4.5, -2.5)

Find the distance between each pair of points. If necessary, round to the See Problem 3.
nearest tenth.

22. J(2, -1), K(2, 5) 23. L(10, 14), M( -8, 14) 24. N( -1, -11), P( -1, -3)

25. A(0, 3), B(0, 12) 26. C(12, 6), D( -8, 18) 27. E(6, -2), F( -2, 4)

28. Q(12, -12), T(5, 12) 29. R(0, 5), S(12, 3) 30. X( -3, -4), Y(5, 5)

Maps  For Exercises 31–35, use the map below. Find the distance between the See Problem 4.

cities to the nearest tenth. 12 y Davenport
31. Augusta and Brookline

32. Brookline and Charleston Everett 8 Charleston x
–8 –4 16
33. Brookline and Davenport 4 Brookline
Augusta
34. Everett and Fairfield
0 4 8 12
35. List the cities in the order of least to
greatest distance from Augusta. Fairfield

B Apply Find (a) PQ to the nearest tenth and (b) the coordinates of the midpoint of PQ.

36. P(3, 2), Q(6, 6) 37. P(0, -2), Q(3, 3) 38. P( -4, -2), Q(1, 3)

39. P( -5, 2), Q(0, 4) 40. P( -3, -1), Q(5, -7) 41. P( -5, -3), Q( -3, -5)

42. P( -4, -5), Q( -1, 1) 43. P(2, 3), Q(4, -2) 44. P(4, 2), Q(3, 0)

45. Think About a Plan  An airplane at T(80, 20) needs to fly to both U(20, 60) and

V(110, 85). What is the shortest possible distance for the trip? Explain.
• What type of information do you need to find the shortest distance?
• How can you use a diagram to help you?

54 Chapter 1  Tools of Geometry

46. Reasoning  The endpoints of AB are A( - 2, - 3) and B(3, 2). Point C lies on AB and
2
is 5 of the way from A to B. What are the coordinates of Point C? Explain how you

found your answer.

47. Do you use the Midpoint Formula or the Distance Formula to find the following?
a. Given points K and P, find the distance from K to the midpoint of KP.
b. Given point K and the midpoint of KP, find KP.

For each graph, find (a) AB to the nearest tenth and (b) the coordinates of the
midpoint of AB.

48. 6 y B 49. 4 y 50. 3 y
A
x x
Ϫ9 O 6 O 4 8 1 x
A Ϫ6 A Ϫ2 O 2
B
Ϫ6 B Ϫ2

51. Coordinate Geometry  Graph the points A(2, 1), B(6, -1), C(8, 7), and D(4, 9).

Draw parallelogram ABCD, and diagonals AC and BD.

ah.s mFi1n1dgmthsee_m01id0p7o_ti0n0ts97o0f.AaiC and BDh. sm11gmse_0107_t00971.ai hsm11gmse_0107_t00972.ai
b. What appears to be true about the diagonals of a parallelogram?

Travel  The units of the subway map at the right are in miles. Suppose the 6y Jackson
routes between stations are straight. Find the distance you would travel Symphony
between each pair of stations to the nearest tenth of a mile. North

52. Oak Station and Jackson Station 4

53. Central Station and South Station Central

54. Elm Station and Symphony Station Cedar 2

55. Cedar Station and City Plaza Station Ϫ4 Ϫ2 City Plaza x

56. Maple Station is located 6 mi west and 2 mi north of City Plaza. O2 4
What is the distance between Cedar Station and Maple Station?
Oak

Ϫ4 South

Elm Ϫ6

57. Open-Ended  Point H(2, 2) is the midpoint of many segments.
a. Find the coordinates of the endpoints of four noncollinear segments that have

point H as their midpoint.

b. You know that a segment with midpoint H has length 8. How many posshibslem11gmse_0107_t00973.ai
noncollinear segments match this description? Explain.

C Challenge 58. Points P( -4, 6), Q(2, 4), and R are collinear. One of the points is the midpoint of the
segment formed by the other two points.

a. What are the possible coordinates of R?
b. Reasoning  RQ = 1160. Does this information affect your answer to part (a)?

Explain.

Lesson 1-7  Midpoint and Distance in the Coordinate Plane 55

Geometry in 3 Dimensions  You can use three coordinates (x, y, z) to locate G z
points in three dimensions.
P E
59. Point P has coordinates (6, -3, 9) as shown at the right. Give the F
coordinates of points A, B, C, D, E, F, and G.

Distance in 3 Dimensions  In a three-dimensional coordinate system, you D
can find the distance between two points (x1, y1, z1) and (x2, y2, z2) with A
this extension of the Distance Formula.
C Oy
d = 2(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2 xB

Find the distance between each pair of points to the nearest tenth.

60. P(2, 3, 4), Q( -2, 4, 9) 61. T(0, 12, 15), V( -8, 20, 12)

Standardized Test Prep

SAT/ACT 62. A segment has endpoints (14, -8) and (4, 12). What are the coordinates of hsm11gmse_0107_t00975.ai

its midpoint?

(9, 10) ( -5, 10) (5, -10) (9, 2)

63. Which of these is the first step in constructing a congruent segment?

Draw a ray. Label two points.

Find the midpoint. Measure the segment.

Short 64. T he midpoint of RS is N( -4, 1). One endpoint is S(0, -7).
Response a. What are the coordinates of R?

b. What is the length of RS to the nearest tenth of a unit?

Mixed Review

Use a straightedge and a compass. See Lesson 1-6.
65. Draw AB. Construct PQ so that PQ = 2AB. See Lesson 1-4.
66. Draw an acute ∠RTS. Construct the bisector of ∠RTS.
See p. 886.
Use the diagram at the right. SP 72. 2 mi = ■ ft
67. Name ∠1 two other ways.
68. If m∠PQR = 60, what is m∠RQS? 1
T QR

Get Ready!  To prepare for Lesson 1-8, do Exercises 69–72.

Complete each statement. Use the conversion table on page 837.
69. 130 in. = ■ ft 70. 14 yd = ■ in. hsm11gms7e_10. 12077_ftt0=09■77y.dai

56 Chapter 1  Tools of Geometry

Concept Byte Partitioning a Line MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Segment
For Use With Lesson 1-7 MG-AGFPSE.9.B1.26. GF-iGndPEth.2e.6p oFinintdotnhea pdoiriencttoendalindeirected
lsiengemseegnmt ebnetwbetewnetewnotwgoivgeinvepnoipnotisntshathtaptapratirttiiotinosns
the seeggmmeennttininaaggiviveennrartaioti.o.

MP 8

The directed line segment from A to B starts at point A and ends at point B.

Suppose point P partitions the directed line segment from A to B in the ratio 2 to 3.
You can think of AB as being divided into 5 congruent parts so that the length of AP is
the length of 2 parts and PB is the length of 3 parts.

Example

LM is the directed line segment from L(−4, 1) to M(5, −5). What are the
coordinates of the point that partitions the segment in the ratio 2 to 1?

Step 1 Graph LM in the coordinate plane. 4y
L2
Step 2 As you move from L to M, you move down 6 units and right
9 units. Divide each of these distances into three equal parts.

vertical distance: 6 , 3 = 2 −6 −4 −2 O 2 x
horizontal distance: 9 , 3 = 3 46
6 units −2
M
Step 3 Beginning at L, move down 2 units and right 3 units to the point −4
at ( - 1, - 1). Repeat this process two more times to reach the −6
points at (2, - 3) and M(5, - 5).

The points at ( - 1, - 1) and (2, - 3) divide LM into three congruent parts. 9 units
Let P be the point with coordinates (2, - 3). Then the ratio LP to PM is 2
to 1. So, the coordinates of the point that partitions the directed line segment

LM in the ratio 2 to 1 are (2, - 3).

Exercises

1. Reasoning  In the Example, how can you show that the points at ( - 1, - 1) and
(2, - 3) divide LM into three congruent parts?

2. RS is the directed line segment from R( - 2, - 3) to S(8, 2). What are the
coordinates of the point that partitions the segment in the ratio 2 to 3?

3. Point C lies on the directed line segment from A(5, 16) to B( - 1, 2) and partitions
the segment in the ratio 1 to 2. What are the coordinates of C?

4. The endpoints of XY are X(2, - 6) and Y( - 6, 2). What are the coordinates of
3
point P on XY such that XP is 4 of the distance from X to Y?

Concept Byte  Partitioning a Line Segment 57

Review Classifying Polygons CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 1-8 Prepares for MG-AMFGS.9A1.12 .GU-sMe gGe.o1m.1e tUricse
sgheaopmees,trtihcesihr ampeas,suthresir, amnedasthuereirs,paronpdetrhtieeisr to
dperospcreirbteieosbtjoecdtess. cribe objects.

In geometry, a figure that lies in a plane is called a plane figure. A polygon is a closed
plane figure formed by three or more segments. Each segment intersects exactly two
other segments at their endpoints. No two segments with a common endpoint are
collinear. Each segment is called a side. Each endpoint of a side is a vertex.

B B B

A A A
C C C

ED ED DE
ABCDE is a polygon. ABCDE is not a polygon. ABCDE is not a polygon.

You can classify a polygon by its number of sides: triangle (3 sides), quadrilateral

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You can also classify a polygon as concave or convex, using the diagonals of the
polygon. A diagonal is a segment that connects two nonconsecutive vertices.

A D MP
RY W KG Q

T S

A convex polygon has no diagonal A concave polygon has at least one
with points outside the polygon. diagonal with points outside the polygon.

In this textbook, a polygon is convex unless otherwise stated. hsm11gmse_0108a_t01056.ai

hsm11gmse_0108a_t01055.ai

Exercises

Is the figure a polygon? If not, explain why.

1. 2. 3. 4.

Classify the polygon by its number of sides. Tell whether the polygon is convex or concave.

5. 6. 7.

hsm11gmse_0108a_t01061h.asmi 11gmse_0108a_t01062h.sami 11gmse_0108a_t01063h.asim11gmse_0108a_t01065.ai

58 Review  Classifying Polygons hsm11gmse_0108a_t01073.ai
hsm11gmse_0108a_t01070.ai

1-8 Perimeter, Circumference, CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Area
MN-AQF.AS.911  2U.sNe-uQn.i1ts.1a sUasewuanyitso ausnadewrsatyantod understand
problems and to guide the solution of multi-step
problems; choose and interpret units consistently
in formulas . . .

MP 1, MP 3, MP 4, MP 7

Objectives To find the perimeter or circumference of basic shapes
To find the area of basic shapes

You and your friend have two choices for a wall
decoration. You say the decoration on the top
will use more wall space. Your friend says the
two decorations will use the same amount of
wall space. Who is correct? Explain.

Think about what
“wall space” means.

MATHEMATICAL

PRACTICES In the Solve It, you considered various ideas of what it means to take up space on a
flat surface.

Lesson Essential Understanding  Perimeter and area are two different ways of
measuring geometric figures.
Vocabulary
• perimeter The perimeter P of a polygon is the sum of the lengths of its sides. The area A
• area of a polygon is the number of square units it encloses. For figures such as
squares, rectangles, triangles, and circles, you can use formulas for
perimeter (or circumference C for circles) and area.

Key Concept  Perimeter, Circumference, and Area

Square Triangle
side length s
s side lengths a, b, and c, ah c
P = 4s base b, and height h b
A = s2
P =a+b+c
A = 12bh

Rectangle Circle hsm1r1gmse_0108_t0
base b and height h
hsm11gmse_0108_rta0d1iu1s8r7a.anid diameter d d
P = 2b + 2h, or
2(b + h) h C = pd, or C = 2pr C
A = bh A = pr2

b

Lesson 1-8  Perimeter, Circumference, and Areah sm11gmse_051098_t01

hsm11gmse_0108_t01188.ai

The units of measurement for perimeter and circumference include inches, feet,
yards, miles, centimeters, and meters. When measuring area, use square units such
as square inches (in.2), square feet (ft2), square yards (yd2), square miles (mi2),
square centimeters (cm2), and square meters (m2).

Problem 1 Finding the Perimeter of a Rectangle

Landscaping  The botany club members are designing a rectangular 4 ft
garden for the courtyard of your school. They plan to place edging on 22 ft
the outside of the path. How much edging material will they need?
16 ft
Step 1 Find the dimensions of the garden, including the path.

Why should you draw For a rectangle, “length” and
a diagram?
A diagram can help you 4 “width” are sometimes used in
see the larger rectangle
formed by the garden 4 16 4 place of “base” and “height.”
and the path, and which
lengths to add together. Width of the garden and path
22  = 4 + 16 + 4 = 24

Length of the garden and path

4  = 4 + 22 + 4 = 30

Step 2 FinHdStMh1e1pGeMrSimE_e0t1e0r8o_fat0h0e45g9arden including the path.
P2Dnu=drkp2eabss+021h-06-09 Use the formula for the perimeter of a rectangle.
= 2(24) + 2(30) Substitute 24 for b and 30 for h.

= 48 + 60 Simplify.

= 108

You will need 108 ft of edging material.

Got It? 1. You want to frame a picture that is 5 in. by 7 in. with a 1-in.-wide frame.
a. What is the perimeter of the picture?
b. What is the perimeter of the outside edge of the frame?

You can name a circle with the symbol }. For example, the circle with center A is
written }A.

The formulas for a circle involve the special number pi (p). Pi is the ratio of any circle’s
circumference to its diameter. Since p is an irrational number,

p = 3.1415926 . . . ,

you cannot write it as a terminating decimal. For an approximate answer, you can use

3.14 or 22 for p. You can also use the key on your calculator to get a rounded decimal
7
for p. For an exact answer, leave the result in terms of p.

60 Chapter 1  Tools of Geometry