Geometry

12. Parking  Two workers paint lines for angled 1 See Problem 3.
parking spaces. One worker paints a line so
that m∠1 = 65. The other worker paints a 23
line so that m∠2 = 65. Are their lines parallel?
Explain. HSM11GMSE_0303_a00839 See Problem 4.

Algebra  Find the value of x for which O } m. 14. 1st pass 08-28-08
13. Durke
95Њ
ഞ 55Њ
(2x Ϫ 5)Њ m
m (x ϩ 25)Њ ഞ

15. 16. ഞ m
(3x Ϫ 33)Њ
hs1m05Њ11gmse_0303_t00381.ai
hsmm 11gmse_(02x3ϩ032_6)tЊ00380.ai
(3x Ϫ 18)Њ
ഞ

B Apply Developing Proof  Use the given information to determine which lines,

if any, are parallel. Justify each conclusion with a theorem or postulate. 15 9 11 a
10 12
17. ∠h2sims s1u1ppglmemseen_t0a3ry0t3o_∠t030. 382.ai 18. ∠hs1m≅1∠1g3 mse_0303_t003823.6ai
b
19. ∠6 is supplementary to ∠7. 20. ∠9 ≅ ∠12 37
21. m∠7 = 65, m∠9 = 115 22. ∠2 ≅ ∠10 48 m

23. ∠1 ≅ ∠8 24. ∠8 ≅ ∠6 ഞ

25. ∠11 ≅ ∠7 26. ∠5 ≅ ∠10

Algebra  Find the value of x for which O } m. 28. hsm11gmse_0303_t00384.ai
27. ഞ m 5x Њ
ഞ
27x Њ 2x Њ m
19xЊ 17xЊ
(5x ϩ 40)Њ

29. Write a paragraph proof. ഞ2
3
Proo f GPhrisovmevne1::  1/∠g}2mmissesu_p0p3l0em3_etn0ta0r3y8t5o.∠ai7.
hsm11gmse_0303_t0m0386.a6i

7

Lesson 3-3  Proving Lines Parallel hsm11gmse_0303_t01063187.ai

30. Think About a Plan  If the rowing crew at the right
strokes in unison, the oars sweep out angles of equal
measure. Explain why the oars on each side of the
shell stay parallel.

• What type of information do you need to prove
lines parallel?

• How do the positions of the angles of equal
measure relate?

Algebra  Determine the value of x for which r } s.
Then find mj1 and mj2.

31. m∠1 = 80 - x, m∠2 = 90 - 2x r1
32. m∠1 = 60 - 2x, m∠2 = 70 - 4x s2
33. m∠1 = 40 - 4x, m∠2 = 50 - 8x

34. m∠1 = 20 - 8x, m∠2 = 30 - 16x

Use the diagram at the right below for Exercises 35–41.

Open-Ended  Use the given information. Stahtseman1o1tghmersfeac_t0a3b0o3u_t to0n0e3o8f 8th.aei

given angles that will guarantee two lines are parallel. Tell which lines will be
parallel and why.

35. ∠1 ≅ ∠3 36. m∠8 = 110, m∠9 = 70 k

37. ∠5 ≅ ∠11 38. ∠11 and ∠12 are supplementary. j 1
42
3
39. Reasoning  If ∠1 ≅ ∠7, what theorem or postulate can you use to
show that / } n? 95
12 10 8 6

Write a flow proof. 11 7

40. Given:  / } n, ∠12 ≅ ∠8 41. Given:  j } k, m∠8 + m∠9 = 180 ᐍ
Proo f Prove:  j } k P roo f Prove:  / } n n

C Challenge Which sides of quadrilateral PLAN must be parallel? Explain.

42. m∠P = 72, m∠L = 108, m∠A = 72, m∠N = 108 hsm11gmse_0303_t00389.ai
43. m∠P = 59, m∠L = 37, m∠A = 143, m∠N = 121

44. m∠P = 67, m∠L = 120, m∠A = 73, m∠N = 100
45. m∠P = 56, m∠L = 124, m∠A = 124, m∠N = 56

46. Write a two-column proof to prove the following: If a transversal intersects two
Proof parallel lines, then the bisectors of two corresponding angles are parallel. (Hint:

Start by drawing and marking a diagram.)

162 Chapter 3  Parallel and Perpendicular Lines

Standardized Test Prep

SAT/ACT Use the diagram for Exercises 47 and 48.

47. For what value of x is c } d? 43 a b
21 53 136Њ c

23 1 (x ϩ 21)Њ
d
48. If c } d, what is m∠1? 136
24 146 2x Њ

44

49. Which of the following is always a valid conclusion for the hypothesis?

If two angles are congruent, then ? . hsm11gmse_0303_t00390.ai
they are right angles
they have the same measure

Short they share a vertex they are acute angles
Response
50. What is the value of x in the diagram at the right? 2xЊ (8x Ϫ 10)Њ
1.6 17
10 19

51. Draw a pentagon. Is your pentagon convex or concave? Explain.

Mixed Review hsm11gmse_0303_t00391.ai

Find mj1 and mj2. Justify each answer. 53. See Lesson 3-2.
52. W X 2C
A 94Њ
110Њ B 1
D
21 66Њ
ZY

Get Ready!  To prepare for Lesson 3-4, do Exercises 54–57.

Deterhmsimne1w1hgemthseer_e0a3ch03st_att0em03e9n2t i.as ialways, sometimes, or never true. See Lessons 1-6 and 3-1.
hsm11gmse_0303_t00393.ai
54. Perpendicular lines meet at right angles.

55. Two lines in intersecting planes are perpendicular.

56. Two lines in the same plane are parallel.

57. Two lines in parallel planes are perpendicular.

Lesson 3-3  Proving Lines Parallel 163

3-4 Parallel and MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Perpendicular Lines
MG-AMFGS.9A1.32 .GA-pMplGy .g1e.o3m  Aetprpiclymgeetohmodestrtioc msoelvtheoddesstigon
psorolvbeledmes.ign problems.

MP 1, MP 3, MP 4

Objective To relate parallel and perpendicular lines

Look at the given Jude and Jasmine leave school together to walk home. Then Jasmine
angle markings. cuts down a path from Schoolhouse Road to get to Oak Street and
What do they Jude cuts down another path to get to Court Road. Below is a diagram
tell you? of the route each follows home. What conjecture can you make about
Oak Street and Court Road? Explain.
MATHEMATICAL
Oak Street
PRACTICES
School Schoolhouse Road

Court Road

In the Solve It, you likely made your conjecture about Oak Street and Court Road based
on their relationships to Schoolhouse Road. In this lesson, you will use similar reasoning
to prove that lines are parallel or perpendicular.

Essential Understanding  You can use the relationsh4HitSphMsp1oa1fsGstwM1o2S-El0i_7n0e3s04to_aa0t0h8i4r0d
line to decide whether the two lines are parallel or perpendi-c0u8lar to each other.

Durke

Theorem 3-8 If . . . Then . . .
a } b and b } c a}c
Theorem a
If two lines are parallel to
the same line, then they are b
parallel to each other.
c

You will prove Theorem 3-8 in Exercise 7.

hsm11gmse_0304_t00256

164 Chapter 3  Parallel and Perpendicular Lines

Theorem 3-9 If . . . Then . . .
m # t and n # t m}n
Theorem
In a plane, if two lines are t
perpendicular to the same
line, then they are parallel to m
each other.
n

Notice that Theorem 3-9 includes the phrase in a plane. In Exercise 17, you will

consider why this phrase is necessaryh. sm11gmse_0304_t00257

Proof Proof of Theorem 3-9 r st
2
Given: In a plane, r # t and s # t.
1
Prove: r } s
Proof: ∠1 and ∠2 are right angles by the definition of

perpendicular. So, ∠1 ≅ ∠2. Since corresponding angles

are congruent, r } s.

hsm11gmse_0304_t00258

Problem 1 Solving a Problem With Parallel Lines STEM

Carpentry  A carpenter plans to install molding on the top

sides and the top of a doorway. The carpenter cuts the 45Њ 45Њ
45Њ
ends of the top piece and one end of each of the side side 45Њ side
pieces at 45° angles as shown. Will the side pieces of
molding be parallel? Explain.

The angles at the Determine whether Visualize fitting the pieces together to
connecting ends are 45°. the side pieces of
molding are parallel. tfohremnenwewanagnlgelsesth.oUsdsmeecii1dnef1owrgmhmaettihsoeenr _athb0eo3ut04_t00259

sides are parallel.

Yes, the sides are parallel. When the pieces fit together, they form 45° + 45°, or 90°,
angles. So, each side is perpendicular to the top. If two lines (the sides) are

perpendicular to the same line (the top), then they are parallel to each other.

Got It? 1. Can you assemble the pieces at the right to 60Њ 60Њ 30˚
form a picture frame with opposite sides 30˚

parallel? Explain. 60Њ 60Њ

Lesson 3-4  Parallel and Perhpsemnd1ic1uglamr Lsien_e0s 304_t00260 165

Theorems 3-8 and 3-9 give conditions that allow you to conclude that lines are parallel.
The Perpendicular Transversal Theorem below provides a way for you to conclude that
lines are perpendicular.

Theorem 3-10  Perpendicular Transversal Theorem

Theorem If . . . Then . . .
In a plane, if a line is n # / and / } m n#m
perpendicular to one of two
parallel lines, then it is also n ᐍ
perpendicular to the other.

m

You will prove Theorem 3-10 in Exercise 10.

sbTpoheelreiipdnPe,aen<ArdppClie>acanunndleadi.crT<uBtholDae<>BrdaDTirar>e.agIpnrnaasmfrvaaeclarltse,tal<tE.lhC<ETe>Chra>eingiosdhrpet<hBmesDshrpm>osetawa1nrtsed1ewssigckhtmuehylwaa.stIrebntt_heotce0h<aA3eluiCn0rs>,ee4ebcs_ttuhmatte0nuiyg0tsuia2tsrl6aenr1ot C A B
G D

H

not in the same plane. EF

Proof Problem 2 Proving a Relationship Between Two Lines hsm11gmse_0304_t00262

Which line has a Given: In a plane, c # b, b # d, and d # a. c
relationship to both Prove: c # a
lines c and d?
You know c # b and Proof: Lines c and d are both perpendicular to line b, so c } d d
b # d. So, line b relates because two lines perpendicular to the same line are b
to both lines c and d. parallel. It is given that d # a. Therefore, c # a because a
a line that is perpendicular to one of two parallel lines

is also perpendicular to the other (Perpendicular

Transversal Theorem).

Got It? 2. In Problem 2, could you also conclude a } b? Explain. hsm11gmse_0304_t00263

166 Chapter 3  Parallel and Perpendicular Lines

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. Main Street intersects Avenue A and Avenue B.
3. Explain why the phrase in a plane is not necessary in
Avenue A is parallel to Avenue B. Avenue A is also
perpendicular to Main Street. How are Avenue B and Theorem 3-8.
Main Street related? Explain.
4. Which theorem or postulate from earlier in the
2. In the diagram below, lines a, b, and c are coplanar. chapter supports the conclusion in Theorem 3-9? In
What conclusion can you make about lines a and b? the Perpendicular Transversal Theorem? Explain.
Explain.
ca 5. Error Analysis  Shiro sketched mn
b coplanar lines m, n, and r on r
his homework paper. He claims

that it shows that lines m and

n are parallel. What other

information do you need about

line r in order for Shiro’s claim to be true? Explain.

hsm11gmse_0304_t0039

Practichesma1n1dgmPsreo_0b3l0e4m_t-0S03o9l4v.aini g Exercises MATHEMATICAL

PRACTICES

A Practice 6. A carpenter is building a trellis for vines to grow See Problem 1.

on. The completed trellis will have two sets of 12 3
AB
diagonal pieces of wood that overlap each other.
C
a. If pieces A, B, and C must be parallel, what
must be true of ∠1, ∠2, and ∠3? D

b. The carpenter attaches piece D so that it is

perpendicular to piece A. If your answer to

part (a) is true, is piece D perpendicular to

pieces B and C? Justify your answer.

7. Developing Proof  Copy and complete this paragraph proof of Theorem 3-8 See Problem 2.

for three coplanar lines. hsm11gmse_0304_t00396.ai 1
Given:  / } k and m } k

Prove:  / } m k 2
Proof:  Since / } k, ∠2 ≅ ∠1 by the a. ? Theorem. Since m } k, ᐍ 3
m
b. ? ≅ c. ? for the same reason. By the Transitive

Property of Congruence, ∠2 ≅ ∠3. By the d. ?

Theorem, / } m.

8. Write a paragraph proof. c d
Proo f Given:  In a plane, a # b, b # c, and c } d. b

Prove:  a } d hsm11gmse_0304_t00397.ai

a

Lesson 3-4  Parallel and Perpehndsmicu1la1rgLminesse _0304_t00398.a1i 67

B Apply 9. Think About a Plan  One traditional type of log cabin is a single rectangular room. a b
Suppose you begin building a log cabin by placing four logs in the shape of a c
rectangle. What should you measure to guarantee that the logs on opposite walls
are parallel? Explain.
• What type of information do you need to prove lines parallel?
• How can you use a diagram to help you?
• What do you know about the angles of the geometric shape?

10. Prove the Perpendicular Transversal Theorem (Theorem 3-10): In a
Proof plane, if a line is perpendicular to one of two parallel lines, then it is
also perpendicular to the other.

Given:  In a plane, a # b and b } c.
Prove:  a # c

The following statements describe a ladder. Based only on the statement, make
a conclusion about the rungs, one side, or both sides of the ladder. Explain.

11. The rungs are each perpendicular to one side. hsm11gmse_0304_t00399.ai

12. The rungs are parallel and the top rung is perpendicular to one side.

13. The sides are parallel. The rungs are perpendicular to one side.

14. Each side is perpendicular to the top rung.

15. The rungs are perpendicular to one side. The sides are not parallel.

16. Public Transportation  The map at the right is a section of a Salto del Isabel la
subway map. The yellow line is perpendicular to the brown line, Agua Catolica
the brown line is perpendicular to the blue line, and the blue line ,
is perpendicular to the pink line. What conclusion can you make ,
about the yellow line and the pink line? Explain. Ni~nos Doctores
Heroes Obrera
17. Writing  Theorem 3-8 states that in a plane, two lines Hospital
perpendicular to the same line are parallel. Explain why the General
phrase in a plane is needed. (Hint: Refer to a rectangular solid
to help you visualize the situation.) Centro Lazaro
Medico Cardenas
,
,

18. Quilting  You plan to sew two triangles of fabric together to make a square for a

quilting project. The triangles are both right triangles and have the same side and

angle measures. What must also be true about the triangles in order to guarantee HSM11GMSE_0304_a00841
3rd pass 12-06-08
that the opposite sides of the fabric square are parallel? Explain.

C Challenge For Exercises 19–24, a, b, c, and d are distinct lines in the same plane. For each Durke

combination of relationships, tell how a and d relate. Justify your answer.

19. a } b, b } c, c } d 20. a } b, b } c, c # d
21. a } b, b # c, c } d 22. a # b, b } c, c } d

23. a } b, b # c, c # d 24. a # b, b } c, c # d

168 Chapter 3  Parallel and Perpendicular Lines

25. Reasoning  Review the reflexive, symmetric, and transitive properties for
congruence in Lesson 2-5. Write reflexive, symmetric, and transitive statements for
“is parallel to” (}). Tell whether each statement is true or false. Justify your answer.

26. Reasoning  Repeat Exercise 25 for “is perpendicular to” ().

Standardized Test Prep

SAT/ACT 27. In a plane, line e is parallel to line f, line f is parallel to line g, and line h is
perpendicular to line e. Which of the following MUST be true?

e } g h } f g } h e } h

28. Which point lies nearest to (5, 2) in the coordinate plane?

(1, 3) (0, 2) (4, 5) (4, 10)
Short
29. Which of the following is NOT a reason for proving two lines parallel.
Response
The lines are both # to the same line. Vertical angles are congruent.

Corresponding angles are congruent. The lines are both } to the same line.

30. The diameter of a circle is the same length as the side of a square. The perimeter of
the square is 16 cm. Find the diameter of the circle. Then find the circumference of
the circle in terms of p.

Mixed Review

Algebra  Determine the value of x for which a } b. See Lesson 3-3.
31. a
32. ab
124Њ (3x Ϫ 2)Њ

44Њ

b (2x ϩ 18)Њ

Use a protractor. Classify each angle as acute, right, or obtuse. See Lesson 1-4.

33. 34. 35.
hsm11gmse_0304_t00401.ai
hsm11gmse_0304_t00402.ai

Get Ready!  To prepare for Lesson 3-5, do Exercises 36–39.

Solve each equation. hsm11gmse_0304_t00404.ai See p. 886.

36. 3h0s+m9101+gxm=se1_800 304_t00403.ai 37. 55 + x + 105 = 180hsm11gmse_0304_t00400.ai

38. x + 50 = 90 39. 32 + x = 90

Lesson 3-4  Parallel and Perpendicular Lines 169

Concept Byte Perpendicular Lines CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Planes
Use With Lesson 3-4 Extends GM-ACFOS.A.9.112  .KGn-oCwO.p1r.e1c isKendoewfinpirteiocnisseof
adnefginlei,ticoirncsleo,fpaenrpgelen,dciicruclea,rplienrep,epnadriaclulelalrlilninee, ,and
activity lpinaeraslleeglmlineen,ta.n.d. line segment . . .

MP 7

As you saw in Chapter 1, you can use a polygon to represent a plane in space. You
can sketch overlapping polygons to suggest how two perpendicular planes intersect
in a line.


Draw perpendicular planes A and B intersecting in <CD>.
Step 1 Draw plane A and <CD> in plane A.
Step 2 Draw two segments that are perpendicular to <CD>. One segment should pass

through point C. The other segment should pass through point D. The
segments represent two lines in plane B that are perpendicular to plane A.

Step 3 Connect the segment endpoints to draw plane B. Plane B is perpendicular
to plane A because plane B contains lines perpendicular to plane A.

D D B
A C D

C A A

C

Exhesmrc1i1sgemsse_0304fb_t01453.ai hsm11gmse_0304fb_t01455.ai

1. Draw a plane in tshpiardcel.inTehethnadt riaswpetrwhposemnlidn1iec1sugtlhamarttsoaere_ea0icn3h0toh4feftbphle_attnw0e1oa4nli5nd4ei.snataietrpseocint taAt .
point A. Draw a

What is the relationship between the third line and the plane?

2. a. Draw a plane and a point in the plane. Draw a line perpendicular to the plane
at that point. Can you draw more than one perpendicular line?

b. Draw a line and a point on the line. Draw a plane that is perpendicular to the
line at that point. Can you draw more than one perpendicular plane?

3. Draw two planes perpendicular to the same line. What is the relationship between
the planes?

4. Draw line / through plane P at point A, so that line / is perpendicular to plane P.
a. Draw line m perpendicular to line / at point A. How do m and plane P relate?

Does this relationship hold true for every line perpendicular to line / at point A?
b. Draw a plane Q that contains line /. How do planes P and Q relate? Does this

relationship hold true for every plane Q that contains line /?

170 Concept Byte  Perpendicular Lines and Planes

3-5 Parallel Lines and CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Triangles
MG-ACFOS.C.9.120. GP-rCovOe.3th.1e0o rePmrosvaebtohuetotrreimansgalebsou. .t .triangles;
umseeatshueroesreomf sinatbeoriuotr tarniagnlegsleosftao tsroialvneglperosublmemtos.1.8.0. °.
mMePas1u,rMesPof3i,nMtePrio6r angles of a triangle sum to 180°.
MP 1, MP 3, MP 6

Objectives To use parallel lines to prove a theorem about triangles
To find measures of angles of triangles

Can you use your Draw and cut out a large triangle. What is the sum of the angle
result to make a measures of the triangle? Explain. Do not use a protractor. (Hint: Tear
conjecture about off and rearrange the three corners of the triangle.)
other triangles?
2
MATHEMATICAL
13
PRACTICES

In the Solve It, you may have discovered that you can rearrange the corners of the
triangle to form a straight angle. You can do this for any triangle.

Essential Understanding  The sum of the angle measures of a triangle is always

the same. HSM11_GMSE_0305_a_00842

Lesson The Solve It suggests an important theorem about 4Dtrtuiharknpeagslses1.2T-0o6p-0r8ove this theorem, you
will need to use parallel lines.
Vocabulary
• auxiliary line Postulate 3-2  Parallel Postulate
• exterior angle of Through a point not on a line, there is one and only one line parallel to the given line.

a polygon P
• remote interior
ᐍ
angles

There is exactly one line through P parallel to /.

hsm11gmse_0305_t00264

Lesson 3-5  Parallel Lines and Triangles 171

Theorem 3-11  Triangle Angle-Sum Theorem

The sum of the measures of the angles of a triangle is 180.
B

AC
m∠A + m∠B + m∠C = 180

hsm11gmse_0305_t10399.ai

The proof of the Triangle Angle-Sum Theorem requires an auxiliary line. An auxiliary
line is a line that you add to a diagram to help explain relationships in proofs. The red
line in the diagram below is an auxiliary line.

Proof Proof of Theorem 3-11: Triangle Angle-Sum Theorem

Given: △ABC PB R
Prove: m∠A + m∠2 + m∠C = 180 12 3

AC

Statements Reasons
1) Draw <PR> through B, parallel to AC.
1) Parallel Phossmtul1a1tegmse_0305_t00266
2) ∠PBC and ∠3 are supplementary.
2) ⦞ that form a linear pair are suppl.
3) m∠PBC + m∠3 = 180 3) Definition of suppl. ⦞
4) m∠PBC = m∠1 + m∠2 4) Angle Addition Postulate
5) m∠1 + m∠2 + m∠3 = 180 5) Substitution Property
6) ∠1 ≅ ∠A and ∠3 ≅ ∠C 6) If lines are }, then alternate interior ⦞ are ≅.
7) Congruent ⦞ have equal measure.
7) m∠1 = m∠A and m∠3 = m∠C 8) Substitution Property
8) m∠A + m∠2 + m∠C = 180

When you know the measures of two angles of a triangle, you can use the Triangle
Angle-Sum Theorem to find the measure of the third angle.

172 Chapter 3  Parallel and Perpendicular Lines

Problem 1 Using the Triangle Angle-Sum Theorem B
Algebra  What are the values of x and y in the diagram at the right? 43Њ 49Њ

Which variable A xЊ yЊ zЊ C
should you solve 59Њ D
for first?
Use the Triangle Angle-Sum Theorem 59 + 43 + x = 180 hsm11gmse_0305_t00267
From the diagram, to write an equation involving x.
102 + x = 180
you know two angle Solve for x by simplifying and then x = 78
measures in △ADB. The subtracting 102 from each side. mjADB + mjCDB = 180
third angle is labeled
x°. So use what you ∠ADB and ∠CDB form a linear
know about the angle pair, so they are supplementary.

measures in a triangle to

solve for x first.

Substitute 78 for m∠ADB and y for   x + y = 180
m∠CDB in the above equation. 78 + y = 180

Solve for y by subtracting 78 from y = 102
each side.

Got It? 1. Use the diagram in Problem 1. What is the value of z?

An exterior angle of a polygon is an angle formed by a side and an extension of an

adjacent side. For each exterior angle of a triangle, the two nonadjacent interior angles
are its remote interior angles. In each triangle below, ∠1 is an exterior angle and ∠2
and ∠3 are its remote interior angles.

1 2 3
23 13 1

2

The theorem below states the relationship between an exterior angle and its two remote
interior angles.

hsm11gmse_0305_t00276

Theorem 3-12  Triangle Exterior Angle Theorem

The measure of each exterior angle of a triangle equals the sum of the 2
measures of its two remote interior angles. 13

m∠1 = m∠2 + m∠3

You will prove Theorem 3-12 in Exercise 33.

hsm11gmse_0305_t00277

Lesson 3-5  Parallel Lines and Triangles 173

You can use the Triangle Exterior Angle Theorem to find angle measures.

Problem 2 Using the Triangle Exterior Angle Theorem

W hat information A What is the measure of j1?
can you get from the
diagram? m∠1 = 80 + 18 Triangle Exterior Angle Theorem 1
The diagram shows you
which angles are interior m∠1 = 98 Simplify.
or exterior.
80Њ 18Њ

B What is the measure of j2?
59Њ
124 = 59 + m∠2 Triangle Exterior Angle Theorem
65 = m∠2 Subtract 59 from each side. hsm11gmse_0305_t00278

124Њ 2

Got It? 2. Two angles of a triangle measure 53. What is the measure of an exterior
angle at each vertex of the triangle?
hsm11gmse_0305_t00279.ai

Problem 3 Applying the Triangle Theorems

How can you apply Multiple Choice  When radar tracks an object, the
your skills from
Problem 2 here? reflection of signals off the ground can result in
Look at the diagram.
Notice that you have a clutter. Clutter causes the receiver to confuse the
triangle and information
about interior and real object with its reflection, called a ghost. At the
exterior angles.
right, there is a radar receiver at A, an airplane at

B, and the airplane’s ghost at D. What is the B 30؇
value of x?
A
30 70 x؇
50 80

m∠A + m∠B = m∠BCD Triangle Exterior Angle 80؇ C
Theorem

x + 30 = 80 Substitute.

x = 50 Subtract 30 from
each side.

The value of x is 50. The correct answer is B.

Got It? 3. Reasoning  In Problem 3, can you D
find m∠A without using the Triangle

Exterior Angle Theorem? Explain.

174 Chapter 3  Parallel and Perpendicular Lines

HSM11GMSE_0305_a00955

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

Find the measure of the third angle of a triangle given PRACTICES
the measures of two angles.
7. Explain how the Triangle Exterior Angle Theorem

makes sense based on the Triangle Angle-Sum

1. 34 and 88 2. 45 and 90 Theorem.

3. 10 and 102 4. x and 50 8. Error Analysis  The measures of the interior angles
of a triangle are 30, x, and 3x. Which of the following
In a triangle, j1 is an exterior angle and j2 and j3 methods for solving for x is incorrect? Explain.
are its remote interior angles. Find the missing angle
measure. A. B. x + 3x + 30 = 180
x + 3x = 30 4x + 30 = 180
5. m∠2 = 24 and m∠3 = 106 4x = 30 4x = 150
x = 7.5 x = 37.5
6. m∠1 = 70 and m∠2 = 32

hsm11gmse_0305_t00415.ai
MATHEMATICAL hsm11gmse_0305_t00416.ai
Practice and Problem-Solving Exercises
PRACTICES

A Practice Find mj1. See Problem 1.
57Њ
9. 1 117Њ 10. 52.2Њ 11. 33Њ
33Њ
1
44.7Њ 1

Algebra  Find the value of each variable.

12. hsm11gmse_0305_t004081.a3i. hs7m0Њ 11gm30Њse_0305_t004091.a4i. hsm11gmse_0305_t00410.ai

30Њ 40Њ xЊ 30Њ

80Њ xЊ yЊ zЊ yЊ cЊ

Use the diagram at the right for Exercises 15 and 16. See Problem 2.
15. ah. sWmh1ic1hgomf tshee_n0u3m0b5e_rte0d0a4n1g1le.saai rehesxmter1i1org amngslee_s?0305_t00412.ai
4
b. Name the remote interior angles for each exterior angle. 1 hsm11gmse2_03055_t00413.ai

c. How are exterior angles 6 and 8 related?

16. a. How many exterior angles are at each vertex of the triangle? 8 3
b. How many exterior angles does a triangle have in all? 6
7

hsm11gmse_0305_t00414.ai

Lesson 3-5  Parallel Lines and Triangles 175

Algebra  Find each missing angle measure.

17. 18. 128.5Њ 19.
60Њ 2 13Њ 45Њ

1 63Њ 3 4 47Њ

20. A ramp forms the angles shown at thehrsimgh1t.1gmse_0305_t00418.ai See Problem 3.
aЊ bЊ 72Њ
What are the values of a and b?
hsm11gmse_0305_t00419.ai
21. Ahlsomun1g1egchmasireh_a0s3d0if5fe_rte0n0t 4se1t7ti.nagis that change the

angles formed by its parts. Suppose m∠2 = 71 and
m∠3 = 43. Find m∠1.
hsm11gmse_0305_t00420.ai
B Apply
Algebra  Use the given information to find the unknown

angle measures in the triangle. 2

22. The ratio of the angle measures of the acute angles in a 1
right triangle is 1∶2. 3

23. The measure of one angle of a triangle is 40. The
measures of the other two angles are in a ratio of 3∶4.

24. The measure of one angle of a triangle is 108. The
measures of the other two angles are in a ratio of 1∶5.

25. Think About a Plan  The angle measures of △RST are
represented by 2x, x + 14, and x - 38. What are the
angle measures of △RST ?

• How can you use the Triangle Angle-Sum Theorem to

write an equation?
• How can you check your answer?

26. Prove the following theorem: The acute angles of A B
Proof a right triangle are complementary. C

Given:  △ABC with right angle C

Prove:  ∠A and ∠B are complementary.

27. Reasoning  What is the measure of each angle of an equiangular triangle? Explain.

28. Draw a Diagram  Which diagram below correctly represents the following

description? Explain your reasoning. hsm11gmse_0305_t00430.ai

Draw any triangle. Label it △ABC. Extend two sides of the triangle to form two

exterior angles at vertex A.

I. B II. B III. C

AC C BA
A

176 Chapter 3  Parallel ahndsmPe1rp1egnmdisceul_a0r 3Li0n5e_s t01172.ahi sm11gmse_0305_t01174.aihsm11gmse_0305_t01177.ai

Find the values of the variables and the measures of the angles.

29. Q 30. C B
(2x ϩ 4)Њ (8 x Ϫ 1)Њ

(2x Ϫ 9)Њ xЊ (4x ϩ 7)Њ
P R A
32. B
31. eЊ F
E dЊ 32Њ hsmy1Њ 1gmsex_Њ0305_t00421.ai
hsm11gmseH_0cЊ30555Њ_t0ba0ЊЊ423.ai
54Њ zЊ 52Њ
G ADC

33. Prove the Triangle Exterior Angle Theorem (Theorem 3-12).

Proof

sTuhhmesmm o1fe ta1hsgeum rmeseoeafs_eu0arc3ehs0 eo5xf_ tiett0sri0tow4ro2anr8eg.maleiootfeaintrtiearniogrleaneqglueasl.s hthsem11gmse_0305_t002424.ai

Given:  ∠1 is an exterior angle of the triangle. 14 3

Prove:  m∠1 = m∠2 + m∠3

34. Reasoning  Two angles of a triangle measure 64 and 48. What is the measure of the

largest exterior angle of the triangle? Explain. hsm11gmse_0305_t00432.ai

35. Algebra  A right triangle has exterior angles at each of its acute angles with

measures in the ratio 13∶14. Find the measures of the two acute angles of the

right triangle.

C Challenge Probability  In Exercises 36–40, you know only the given information about
the measures of the angles of a triangle. Find the probability that the triangle
is equiangular.

36. Each is a multiple of 30. 37. Each is a multiple of 20.

38. Each is a multiple of 60. 39. Each is a multiple of 12.

40. One angle is obtuse.

41. In the figure at the right, CD # AB and CD bisects A (3x Ϫ 2)Њ
∠ACB. Find m∠DBF . D

42. If the remote interior angles of an exterior angle of
a triangle are congruent, what can you conclude
about the bisector of the exterior angle? Justify your
answer.

FB (5x Ϫ 20)Њ C

hsm11gmse_0305_t00433.ai 177

Lesson 3-5  Parallel Lines and Triangles

Standardized Test Prep

SAT/ACT 43. The measure of one angle of a triangle is 115. The other two angles are congruent.
What is the measure of each of the congruent angles?

32.5 57.5 65 115

44. The center of the circle at the right is at the origin. What is the 4y
approximate length of its diameter?

2 5.6 2 (2, 2)

2.8 8 x

45. One statement in a proof is “∠1 and ∠2 are supplementary angles.” Ϫ4 Ϫ2 O 24

The next statement is “m∠1 + m∠2 = 180.” Which is the best Ϫ2

justification for the second statement based on the first statement? Ϫ4

The sum of the measures of two right angles is 180.

Angles that form a linear pair are supplementary.

Definition of supplementary angles hsm11gmse_0305_t10541.ai
The measure of a straight angle is 180.

Extended 46. △ABC is an obtuse triangle with m∠A = 21 and ∠C is acute.
Response a. What is m∠B + m∠C? Explain.
b. What is the range of whole numbers for m∠C? Explain.

c. What is the range of whole numbers for m∠B? Explain.

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 3
Look back at the blueprint on page 139 of the plan for a city park.

a. Choose three triangles in the blueprint. Use the Triangle Angle-Sum Theorem to
write an equation relating the measures of the angles of each of your triangles.

b. Choose one of the triangles you looked at in part (a). Find the measure of each
angle of your triangle. Show all your steps and explain your reasoning.

178 Chapter 3  Parallel and Perpendicular Lines

Concept Byte Exploring Spherical CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Geometry
Use With Lesson 3-5 Extends GM-ACFOS.A.9.112  .KGn-oCwO.p1r.e1c isKendoewfinpirteiocnisseof
adnefginlei,ticoirncsleo,fpaenrpgelen,dciicruclea,rplienrep,epnadriaclulelalrlilninee, ,and
activity lpinaeraslleeglmlineen,ta.n.d. line segment . . .

MP 8

Euclid was a Greek mathematician who identified many of the Great circle
definitions, postulates, and theorems of high school geometry. Euclidean
geometry is the geometry of flat planes, straight lines, and points.

In spherical geometry, the curved surface of a sphere is studied. A “line”
is a great circle. A great circle is the intersection of a sphere and a plane
that contains the center of the sphere.

1 hsm11gmse_03fa_t00507.ai

You can use latitude and longitude to identify positions on Earth. Look Longitudes Latitudes
at the latitude and longitude markings on the globe.
Equator
1. Think about “slicing” the globe with a plane at each latitude. Do any
of your “slices” contain the center of the globe?

2. Think about “slicing” the globe with a plane at each longitude. Do
any of your “slices” contain the center of the globe?

3. Which latitudes, if any, suggest great circles? Which longitudes, if
any, suggest great circles? Explain.

You learned in Lesson 3–5 that through any point not on a line, there is HSM11_GMSE_03fa_a_00843
one and only one line parallel to the given line (Parallel Postulate). That 3rdd pass 12-06-08
statement is not true in spherical geometry. In spherical geometry, Durke

through a point not on a line, there is no line parallel to the given line.

Since lines are great circles in spherical geometry, two lines always
intersect. In fact, any two lines on a sphere intersect at two points, as
shown at the right.

hsm11gmse_03fa_t00508.ai

Concept Byte  Exploring Spherical Geometry 179

One result of the Parallel Postulate in Euclidean geometry is the Triangle Angle-Sum
Theorem. The spherical geometry Parallel Postulate gives a very different result.

2

Hold a string taut between any two points on a sphere. The string forms a “segment”
that is part of a great circle. Connect three such segments to form a triangle on the
sphere.

Below are examples of triangles on a sphere.

110Њ 97Њ 77Њ 70Њ
108Њ 139Њ

49Њ

4. What is the sum of the angle measures in the first triangle? The second triangle?
The third triangle?

h5.s mHo1w1garme sthee_s0e3refasu_ltt0s 0d5if0fe9re.anit from the Triangle Angle-Sum Theorem in

Euclidean geometry? Explain.

Exercises

For Exercises 6 and 7, draw a sketch to illustrate each property of spherical
geometry. Explain how each property compares to what is true in Euclidean
geometry.

6. There are pairs of points on a sphere through which you can draw more than
one line.

7. Two equiangular triangles can have different angle measures.

8. For each of the following properties of Euclidean geometry, draw a
counterexample to show that the property is not true in spherical geometry.

a. Two lines that are perpendicular to the same line do not intersect.
b. If two angles of one triangle are congruent to two angles of another triangle,

then the third angles are congruent.

9. a. The figure at the right appears to show parallel lines on a sphere. Explain why
this is not the case.

b. Explain why a piece of the top circle in the figure is not a line segment.
(Hint: What must be true of line segments in spherical geometry?)

10. In Euclidean geometry, vertical angles are congruent. Does this seem to be true in
spherical geometry? Explain. Make figures on a globe, ball, or balloon to support
your answer.

hsm11gmse_03fa_t00510.ai

180 Concept Byte  Exploring Spherical Geometry

3 Mid-Chapter Quiz athXM

MathXL® for SchoolOL
R SCHO Go to PowerGeometry.com
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FO

Do you know HOW? F G Find mj1. 16. 30Њ 1
E H 15. 95Њ
Identify the following. Lines
and planes that appear to be 1
parallel are parallel.

1. all segments parallel to HG 100Њ 42Њ
Find the value of x for which a } b.
2. a plane parallel to plane EFB B C
3. all segments skew to EA

4. all segments parallel to A D 17. hsm11gmse(x_ϩ0366m)Њq_t0144ha2s.mai11gmse_03mq_t01443.ai
plane ABCD

(2x Ϫ 8)Њ b

Use the diagram below for Exercises 5–14.

a b c hsm11gmse_03mq_t00502.ai

1 23 d 18.
4 56
88Њ
7 98 e
ahsm11gmse_034xmЊ q_t00504.ai

b

Name two pairs of each angle type. 19. What is the value of x?

5. corresponding angles hAsm11gmse_03mq_t00B 505.ai

6. alternhatsemin1t1ergiomr aseng_l0e3s mq_t00503.ai (2x ϩ 2)Њ

7. same-side interior angles 108Њ
DC
State the theorem or postulate that justifies
each statement. Do you UNDERSTAND?
20. sRkheesawmso?1nE1ixnpgglma  iCnsae.n_a03pamirqo_f tli0n0e5s 0b6e .baoith parallel and
8. ∠7 ≅ ∠9 9. ∠4 ≅ ∠5 21. Open-Ended  Give an example of parallel lines in the

10. m∠1 + m∠2 = 180 real world. Then describe how you could prove that
the lines are parallel.
Complete each statement. 22. Reasoning  Lines /, r, and s are coplanar. Suppose /
11. If ∠5 ≅ ∠9, then ? } ? . is perpendicular to r and r is perpendicular to s. Is /
12. If ∠4 ≅ ? , then d } e. perpendicular to s? Explain.
13. If e # b, then e # ? .
14. If c # d, then b # ? .

Chapter 3  Mid-Chapter Quiz 181

3-6 Constructing Parallel CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Perpendicular Lines
GM-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl gaekoemfoertmricalcognesotmruecttriiocns
wcointhstaruvcatiroientsy wofitthooalsvaarniedtymoefthtooodls .a.n.dcomnestthruocdtsin.g. .
pcoenrpsetrnudcitcinuglarpelirnpeesn.d.ic.ualnadr lcinoenst.ru. c. taindg caolninsetrpuactrianlglela
tlionea pgaivreanllellinteo tahgroivuegnhlainpeotihnrtonuogthoanptohientlineo.tAolnsothe
Glin-eC.OA.lDs.o13MAFS.912.G-CO.4.13

MP 1, MP 3, MP 5

Objective To construct parallel and perpendicular lines

Draw a line m on a sheet of paper. Fold your m
paper so that line m falls on itself. Label your
Check out the fold line n. Fold your paper again so that n
angles formed by falls on itself. Label your new fold line p. How
the folded lines. are m and p related? How do you know?
How are they
related?

MATHEMATICAL In the Solve It, you used paper-folding to construct lines.
PRACTICES

Essential Understanding  You can also use a straightedge and a comHSpMas1s1GtoMSE_0306_a01690
construct parallel and perpendicular lines. 4th pass 12-06-08

In Lesson 3-5, you learned that through a point not on a line, there is a uniqDuuerkleine

parallel to the given line. Problem 1 shows the construction of this line.

Problem 1 Constructing Parallel Lines N
ᐍ
Construct the line parallel to a given line and
through a given point that is not on the line. N

Given:  line / and point N not on / hsm11gmse_0306_t0029ᐍ3.ai
Construct:  line m through N with m } /
Step 1 Label two points H and J on /. Draw <HN>. HJ

What type of angles Step 2 At N, construct ∠1 congruent to ∠NHJ. N1 m
are j1 and jNHJ? Label the new line m.
They are corresponding hsm11gmse_0306_t00295ᐍ .ai
m}/
angles. HJ

Got It? 1. Reasoning  Why must lines / and m be
parallel?

hsm11gmse_0306_t00300.ai

182 Chapter 3  Parallel and Perpendicular Lines

Problem 2 Constructing a Special Quadrilateral

Construct a quadrilateral with one pair of parallel sides of lengths a and b.

How do you know Given:  segments of lengths a and b a
which constructions
to use? Construct: q uadrilateral AbB, aYnZdw<AitZh> <BY> b
Try sketching the final AZ = a, BY =
figure. This can help you }
visualize the construction
steps you will need. hsm11gmse_0306_t00301.ai

You need a pair of parallel sides, B
A
so construct parallel lines as
hsBm11gmse_0306_t00303.ai
you did in Problem 1. Start by
A
drawing a rAayB>wsuitchh endpoint A. is
Then draw that point B hsBm11gmbse_Y0306_t00305.ai

not on the first ray. a
AZ
Construct congruent corresponding
angles to finish your parallel lines.

Now you need sides of lengths
a and b. In Lesson 1-6, you
learned how to construct
congruent segments. Construct
Y and Z so that BY = b and
AZ = a.

Draw YZ. hsBm11gmbse_Y0306_t00308.ai

a
AZ

ABYZ is a quadrilateral with parallel sides of lengths a and b.

Got It? 2. a. D<ABra> w} <CaDse>,g smoethnat.tLAaBbe=l its hsm11gmse_0306_t00311.ai with
m
length m. Construct quadrilateral ABCD
and CD = 2m.
b. Reasoning  Suppose you and a friend both use the steps in Problem 2 to

construct ABYZ independently. Will your quadrilaterals necessarily have

the same angle measures and side lengths? Explain.

Lesson 3-6  Constructing Parallel and Perpendicular Lines 183

Problem 3 Perpendicular at a Point on a Line

Construct the perpendicular to a given line at a given point on the line.

GCoivnesntr: upcot:i n<Ct PP> ownitlhin<CeP/> # / ᐍ
P

Step 1 Construct two points on / that are equidistant ᐍ
from P. Label the points A and B. APB

Why is it important hsm11gmse_0306_t00312.ai
to open your
compass wider? Step 2 Open the compass wider so the opening is greater
If you don’t, you
won’t be able to draw than 1 AB. With the compass tip on A, draw an arc hᐉ sm11gmse_0306_t00313.ai
intersecting arcs above 2
point P. APB
above point P.

Step 3 Without changing the compass setting, place C
the compass point on point B. Draw an arc that
intersects the arc from Step 2. Label the point hᐍAsm11gPmse_0B306_t00314.ai
of intersection C.

Step 4 Draw <CP>.

<CP> # / hsm11Cgmse_0306_t00316.ai

<EF>. <FG> ᐍ

Got It? 3. Use a<FsGt>ra#ig<EhFte> datgpeotoindtrFa.w  Construct so AP B
that

You can also construct a perpendicular line from a point to a line. Thhsismp1er1pgemndsiceu_l0ar306_t00317.ai

line is unique according to the Perpendicular Postulate. You will prove in Chapter 5 that
the shortest path from any point to a line is along this unique perpendicular line.

Postulate 3-3  Perpendicular Postulate
Through a point not on a line, there is one and only one line perpendicular
to the given line.

P

ᐉ

There is exactly one line through P perpendicular to /.

hsm11gmse_0306_t10397.ai

184 Chapter 3  Parallel and Perpendicular Lines

Problem 4 Perpendicular From a Point to a Line

Construct the perpendicular to a given line through a given point not on the line.

Can you use Given:  line <R/ Ga>nwditpho<iRnGt>R#no/t on / R
similar steps for Construct: 
this problem as in ᐍ
Problem 3?
Step 1 Open your compass to a size greater
Yes. Mark two points on
than the distance from R to /. With the R
/ that are equidistant compass on point R, draw an arc thhastm11gmse_0306_t00320.ai
from the given point.
intersects / at two points. Label the ᐍ
Then draw two

intersecting arcs.

points E and F. E F

Step 2 Place the compass point on E and make R
an arc.
hsm11gmse_0306_t00321.ai

ᐍ

EF

Step 3 Keep the same compass setting. With R
the compass tip on F, draw an arc that
intersects the arc from Step 2. Label the hᐍsm11gmse_0306_t00323.ai
point of intersection G.
EF

G

Step 4 Draw <RG>. hsm11gmsRe_0306_t00324.ai
<RG> # /
ᐍ
EF

G

Got It? 4. Draw <CX> and a point Z not on <CX>. Construct <ZB> so that <ZB> # <CX>.
hsm11gmse_0306_t00325.ai

Lesson 3-6  Constructing Parallel and Perpendicular Lines 185

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. Draw a line / and a point P not on the line. Construct 4. In Problem 3, is AC congruent toBC? Explain.

the line through P parallel to line /. 5. Suppose you use a wider compass setting in
2. pDerrapwen<QdRi>caunladratop<oQiRn>taSt Step 1 of Problem 4. Will you construct a different
on the line. Construct the line perpendicular line? Explain.
point S.

3. Draw a line w and a point X not on the line. 6. Compare and Contrast  How are the constructions in
Construct the line perpendicular to line w at point X. Problems 3 and 4 similar? How are they different?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice For Exercises 7–10, draw a figure l<AikBe>. the given one. Then construct the line See Problem 1.
through point J that is parallel to

7. J 8. A

J

AB B

9. 10. A B

hsmJ11gmse_0306_t00479.ai J

AB hsm11gmse_0306_t00480.ai

For Exercises 11–13, draw two segments. Label their lengths a and b. Construct See Problem 2.

a quadrilateral with one pair of parallel sides as described.
11. Thhsemsi1d1esghmavseel_e0n3gt0h6a_ta0n0d4b8. 1.ai
hsm11gmse_0306_t00483.ai

12. The sides have length 2a and b.
13. The sides have length a and 21b.

For Exercises 14 and 15, draw a figure like the given one. Then construct the line See Problem 3.
that is perpendicular to O at point P.

14. P ᐍ 15.
ᐍ

P

hsm11gmse_0306_t00484.ai

hsm11gmse_0306_t00485.ai

186 Chapter 3  Parallel and Perpendicular Lines

For Exercises 16–18, disrpawerapefingduirceulliakrettoh<eRSg>i.ven one. Then construct the line See Problem 4.
through point P that S

16. P 17. P 18. R P

S

RS
R

B Apply 19. Think About a Plan  Draw an acute angle. Construct an angle congruenthtosmyo1u1rgmse_0306_t00489.ai

a•hn sgWmleh1sao1t dtghomaetssteha_esk0twe3to0ch6an_ogftl0teh0se4aa8rne7ga.laletielorhnoasktmeliki1ne1?tegrmiorsean_g0l3es0.6_t00488.ai

• Which construction(s) should you use?

20. Constructions  Construct a square with side length p. p

21. Writing  Explain how to use the Converse of the Alternate
Interior Angles Theorem to construct a line parallel to the given
line through a point not on the line. (Hint: See Exercise 19.)

hsm11gmse_0306_t00490.ai

For Exercises 22–28, use the segments at the right.

22. Draw a line m. Construct a segment of length b that is perpendicular to line m. a
23. Construct a rectangle with base b and height c. b
24. Construct a square with sides of length a.
25. Construct a rectangle with one side of length a and a diagonal of length b. c

26. a. Construct a quadrilateral with a pair of parallel sides of length c.
b. Make a Conjecture  What appears to be true about the other pair of sides in

the quadrilateral you constructed?
c. Use a protractor, a ruler, or both to check the conjecture you made in part (b).

27. Construct a right triangle with legs of lengths a and b.

28. a. Construct a triangle with sides of lengths a, b, and c. hsm11gmse_0306_t004
b. Construct the midpoint of each side of the triangle.

c. Form a new triangle by connecting the midpoints.

d. Make a Conjecture  How do the sides of the smaller triangle and the sides of the

larger triangle appear to be related?

e. Use a protractor, ruler, or both to check the conjecture you made in part (d).

29. Constructions  The diagrams below show steps for a parallel line construction.

I. G II. G III. G IV. G
Cᐍ
Cᐍ ഞ Cᐍ

a. List the construction steps in the correct order.
b. For the steps that use a compass, describe the location(s) of the compass point.

hsm11gmse_0306_t0h0s4m9151.agimse_0306_t00h49sm2.a1i1gmse_0306_t00h49sm6.a1i1gmse_0306_t00493

Lesson 3-6  Constructing Parallel and Perpendicular Lines 187

C Challenge Draw DG. Construct a quadrilateral with diagonals that are congruent to DG,
bisect each other, and meet the given conditions. Describe the figure.

30. The diagonals are not perpendicular. 31. The diagonals are perpendicular.

Construct a rectangle with side lengths a and b that meets the given condition.

32. b = 2a 33. b = 21a 34. b = 13a 35. b = 23a

Construct a triangle with side lengths a, b, and c that meets the given
conditions. If such a triangle is not possible, explain.

36. a = b = c 37. a = b = 2c 38. a = 2b = 2c 39. a = b + c

Standardized Test Prep

SAT/ACT 40. The diagram at the right shows the construction of <CP> perpendicular to

line / <CthB>ro}u<AgBh> point P. Which of the following must be true? <CB> C
<AC> }
ᐍ
CP = 1 AB AC ≅ BC AP B
2

41. Suppose you construct lines /, m, and n so that / # m and / } n. Which of
the following is true?

m } n m } / n # / nh#smm11gmse_0306_t00494.ai
Short
42. For any two points, you can draw one segment. For any three noncollinear points,
Response
you can draw three segments. For any four noncollinear points, you can draw six

segments. How many segments can you draw for eight noncollinear points? Explain

your reasoning.



Mixed Review See Lesson 3-5.
hsm11gmse_0306_t10513.ai

Find each missing angle measure.

43. 35Њ (y Ϫ 15)Њ 44. (2y Ϫ 1)Њ
xЊ (x Ϫ 28)Њ yЊ
3y Њ

Get Ready!  To prepare for Lesson 3-7, do Exercises 45–47.

Simplhifsymea1c1hgrmatsioe._0305_t00448.ai hsm11gms4e7_.0 13220--55_6 t00449.ai See p. 883.

45. 62 -- (( -- 43)) 46. -12--41

188 Chapter 3  Parallel and Perpendicular Lines

3-7 Equations of Lines in the CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Coordinate Plane
Prepares for GM-AGFPSE..9B1.25. GP-rGovPeEt.h2e.5s loPproevceritheeriaslfoopre
pcraitrearlliealfaonrdpaprearlpleelnadnicduplaerrplienneds i.c.u.lar lines . . .

MP 1, MP 3, MP 4

Objective To graph and write linear equations

Think back! Ski resorts often use steepness to rate the difficulty Difficulty Ratings
What did you of their hills. The steeper the hill, the higher the Easiest
learn in algebra difficulty rating. Below are sketches of three new hills Intermediate
that relates to at a particular resort. Use each rating level only once. Difficult
steepness? Which hill gets which rating? Explain.

MATHEMATICAL A 1190 ft B 1180 ft C 1150 ft
3300 ft 3000 ft 3500 ft
PRACTICES

Lesson The Solve It involves using vertical and horizontal distances to determine steepness.
The steepest hill has the greatest slope. In this lesson you will explore the concept of
Vocabulary slope and how it relates to both the graph and the equation of a line.
• slope
• slope-intercept Essential Understanding  You can graph a line and write its equation when you
know certain facts about the line, such as its slope and a point on the line.
form
• point-slope form HSM11_GMSE_0307_a_00844
3rd pass 10-28-08
Durke

Key Concept  Slope

Definition Symbols Diagram
The slope m of a line is the y
ratio of the vertical change A line contains the run
(rise) to the horizontal change
(run) between any two points. points (x1, y1)and rise (x2, y2)

(x2, y2). y2 - y1 (x1, y1) x
x2 - x1
m = rise = O
run

hsm11gmse_0307_t00280

Lesson 3-7  Equations of Lines in the Coordinate Plane 189

Problem 1 Finding Slopes of Lines

FHww oohhrweisclrohedponein,uyymaoouubfoeckarrnnmsoguwola? A Wmh==at2-4-i5s-1t(h--e24s)lope of line b? c yd a

6 (1, 7) (5, 7)

cy h-coooosred ieniathteerapsoyin2t.’sJust = - 45 b
4 (2, 3)

bp oeisnut’rsext‑ocouosredtihneatseame B What is the slope of line d? (؊1, 2) (4, 0) x
as  x2. Ϫ2 O 6 8
m = 0 - -( -42)
4
= 02 Undefined
Ϫ2 (4, ؊2)

Got It? 1. Use the graph in Problem 1.
a. What is the slope of line a?

b. What is the slope of line c?

hsm11gmse_0307_t00281.ai

As you saw in Problem 1 and Got It 1 the slope of a line can be positive, negative,
zero, or undefined. The sign of the slope tells you whether the line rises or falls
to the right. A slope of zero tells you that the line is horizontal. An undefined slope
tells you that the line is vertical.

yyyy

x x x x
O O O O

Positive slope Negative slope Zero slope Undefined slope

You can graph a line when you know its equation. The equation of a line has different

fhorsmms1. T1wgomfoserm_0s3ar0e7s_hto0w0n28be2low. Recall that the y-intercept of a line is the

y-coordinate of the point where the line crosses the y-axis.

Key Concept  Forms of Linear Equations

Definition Symbols

The slope-intercept form of an equation of y ϭ mx ϩ b
a nonvertical line is y = mx + b, where m slocpe cy-intercept
is the slope and b is the y-intercept.

The point-slope form of an equation of a y Ϫ y1 ϭ m(x Ϫ x1)
nonvertical line is y - y1 = m(x - x1), y-coordinatec slocpe x-ccoordinate
where m is the slope and (x1, y1) is a point
on the line.

190 Chapter 3  Parallel and Perpendicular Lines

Problem 2 Graphing Lines

W hat do you do first? A What is the graph of y = 23x + 1?

D etermine which form of The equation is in slope-intercept form, y = mx + b. The slope m is 2 and the
linear equation you have. 3
y-intercept b is 1.
Then use the equation to
i dentify the slope and a
starting point. Step 1 Graph a point Step 2 Uupse2tuhneistsloapned23r.igGhot
at (0,1).

3 units. Graph a point.

4 y
3

2

x
Ϫ4 O 2

Ϫ2
Step 3 Draw a line through

Ϫ4 the two points.

W hat do you do B What is the graph of y − 3 = −2(x + 3)?

w hen the slope is not The equation is in point-slope form, y - y1 = m(x - x1). The slope m is -2 and a
a fraction? hposmint1(1xg1,my1s)eon_0th3e0l7in_et0is0(2-833, .3a)i.
A rise always needs a
r un. So, write the integer
Step 1 Graph a point
as a fraction with 1 as
at (Ϫ3, 3).
the denominator.

4y

Step 2 Use the slope Ϫ2. Go down 22
2 units and right 1 unit.
Graph a point. 1 x
Ϫ2 O 2

Step 3 Draw a line through
Ϫ4 the two points.

Got It? 2. ba.. GGrraapphh yy -= 23x=--431.(x - 4).
hsm11gmse_0307_t00284.ai

Lesson 3-7  Equations of Lines in the Coordinate Plane 191

You can write an equation of a line when you know its slope and at least one point
on the line.

W hich linear Problem 3 Writing Equations of Lines
equation form should A What is an equation of the line with slope 3 and y-intercept −5?
you use?
When you know the y ϭ mx ϩ b
slope and the y-intercept,
use slope-intercept form. m ϭ 3 b ϭ Ϫ5
When you know the
slope and a point on y ϭ 3x ϩ (Ϫ5) Substitute 3 for m and Ϫ5 for b.
the line, use point-slope
f orm. y ϭ 3x Ϫ 5 Simplify.

B What is an equation of the line through ( −1, 5) with slope 2?
y Ϫ y1 ϭ m(x Ϫ x1)

y1 ϭ 5 m ϭ 2 x1 ϭ Ϫ1

y Ϫ 5 ϭ 2[x Ϫ (Ϫ1)] Substitute (Ϫ1, 5) for (x1, y1) and 2 for m.
y Ϫ 5 ϭ 2(x ϩ 1) Simplify.

Got It? 3. a. What is an equation of the line with slope - 1 and y-intercept 2?
2
b. What is an equation of the line through ( -1, 4) with slope -3?

Postulate 1-1 states that through any two points, there is exactly one line. So, you need
only two points to write the equation of a line.

What is the first thing Problem 4 Using Two Points to Write an Equation y
you need to know? What is an equation of the line at the right? 4 (3, 5)
It doesn’t matter yet
what form of linear Start by finding the slope m = y2 - y1 = 5 - ( - 1) = 6 (؊2, ؊1) x
equation you plan to use. m of the line through the x2 - x1 3 - ( - 2) 5 24
You’ll need the slope given points. Ϫ3 O
for both slope-intercept Ϫ2
form and point-slope You have the slope and
form. you know two points on
the line. Use point-slope
form. y - y1 = m(x - x1)

Use either point for hsm11gmse_0307_t00285.ai
(x1, y1). For example, you
can use (3, 5). y - 5 = 56(x - 3)

192 Chapter 3  Parallel and Perpendicular Lines

Got It? 4. a. What is the equation of the line in Problem 4 if you use ( -2, -1) instead
of (3, 5) in the last step?

b. Rewrite the equations in Problem 4 and part (a) in slope-intercept form

and compare them. What can you conclude?

You know that the slope of a horizontal line is 0 and the slope of a vertical line is
undefined. Thus, horizontal and vertical lines have easily recognized equations.

Problem 5 Writing Equations of Horizontal and Vertical Lines

How is this different What are the equations for the horizontal and vertical lines y
from writing other through (2, 4)? (2, 4)
linear equations?
You don’t need the slope. Every point on the horizontal line through (2, 4) has a y-coordinate 2 x
Just locate the point of 4. The equation of the line is y = 4. It crosses the y-axis at (0, 4). 4
where the line crosses Ϫ2 O
the x-axis (for vertical) or Every point on the vertical line through (2, 4) has an x‑coordinate Ϫ2
y-axis (for horizontal). of 2. The equation of the line is x = 2. It crosses the x-axis at (2, 0).

Got It? 5. a. What are the equations for the horizontal and vertical hsm11gmse_0307_t00286.a
lines through (4, -3)?

b. Reasoning  Can you write the equation of a vertical line

in slope‑intercept form? Explain.

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

For Exercises 1 and 2, find the slope of the line passing PRACTICES
through the given points.
5. Vocabulary  Explain why you think slope-intercept
1. (4, 5) and (6, 15)
form makes sense as a name for y = mx + b. Explain
2. why you think point-slope form make sense as a name

for y - y1 = m(x - x1).

y (2, 5) 6. Compare and Contrast  Graph y = 2x + 5 and
y = -31x + 5. Describe how these lines are alike and
4 how they are different.

2 7. Error Analysis  A classmate m ‫؍‬ 8؊8
10 ؊ (؊2)
x found the slope of the line
O 246 passing through (8, -2) and m‫ ؍‬0
(8, 10), as shown at the right. 12
(5, ؊1)
Describe your classmate’s error.
3. What is an equation of a line with slope 8 and m‫؍‬0
y‑intercept 10? Then find the correct slope of

4. Whshmat1is1agnmeqseu_at0io3n0o7f_at0li1ne40p9as.asiing through the line passing through the

(3, 3) and (4, 7)? given points.

hsm11gmse_0307_t0141

Lesson 3-7  Equations of Lines in the Coordinate Plane 193

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the slope of the line passing through the given points. See Problem 1.

8. y 9. y x
4 6
2 (2, 2) (؊3, 4)
Ϫ4 Ϫ2 O 24 x

(؊1, ؊4) Ϫ4 Ϫ2 O 4

Ϫ2 (3, ؊1)

10. (4, -6), (7, 2) 11. ( -3, 7), ( -1, 4) 12. ( -8, 3), ( -11, 4)

13. ( -6, 2), ( -7, 10) 14. (3, 2), ( -6, 2) 15. (5, 9), (5, -6)

hsm11gmse_0307_t00439.ai hsm11gmse_0307_t00441.ai

Graph each line. See Problem 2.

16. y = x + 2 17. y = 3x + 4
19. y = -53x + 2
18. y = 12x - 1 21. y - 1 = -3(x + 2)
20. y - 3 = 13(x - 3) 23. y + 1 = -23(x + 4)

22. y + 4 = (x - 5)

Use the given information to write an equation of each line. See Problems 3 and 4.

24. slope 3, y-intercept 6 25. slope 21, y-intercept -5
27. slope -3, passes through (4, -1)
26. slope 23, passes through ( -2, -6) 29. y
28. y
(؊2, 6) 6
(3, 5)

(؊5, 3) 2 x (1, 3) x
2
Ϫ6 Ϫ4 Ϫ2 O 246
Ϫ2 Ϫ4 Ϫ2 O 2

30. passes through (0, 5) and (5, 8) 31. passes through (6, 2) and (2, 4)

32. phasssmes1t1hgromugshe_( -034,047)_atn0d0(424, 130.a) i 33. passes through ( -1, 0) and ( -3, -1)

hsm11gmse_0307_t00445.ai
Write the equation of the horizontal and vertical lines though the given point. See Problem 5.

34. (4, 7) 35. (3, -2) 36. (0, -1) 37. (6, 4)

194 Chapter 3  Parallel and Perpendicular Lines

B Apply Graph each line.

38. x = 3 39. y = -2 40. x = 9 41. y = 4

42. Open-Ended  Write equations for three lines that contain the point (5, 6).

43. Think About a Plan  You want to construct
a “funbox” at a local skate park. The skate
park’s safety regulations allow for the ramp on
the funbox to have a maximum slope of 141. If
you use the funbox plan at the right, can you
build the ramp to meet the safety regulations?
Explain.

• What information do you have that you can
use to find the slope?

• How can you compare slopes?

Write each equation in slope-intercept form. 46. -5x + y = 2 47. 3x + 2y = 10
44. y - 5 = 2(x + 2) 45. y + 2 = -(x - 4)

STEM 48. Science  The equation P = - 313d + 1 represents the pressure P in atmospheres a
scuba diver feels d feet below the surface of the water.

a. What is the slope of the line?
b. What does the slope represent in this situation?
c. What is the y-intercept (P-intercept)?
d. What does the y-intercept represent in this situation?

Graph each pair of lines. Then find their point of intersection.

49. y = -4, x = 6 50. x = 0, y = 0 51. x = -1, y = 3 52. y = 5, x = 4

STEM 53. Accessibility  By law, the maximum slope of an access ramp in new construction
is 112. The plan for the new library shows a 3-ft height from the ground to the main
entrance. The distance from the sidewalk to the building is 10 ft. If you assume
the ramp does not have any turns, can you design a ramp that complies with
the law? Explain.

54. a. What is the slope of the x-axis? Explain.
b. Write an equation for the x-axis.

55. a. What is the slope of the y-axis? Explain.
b. Write an equation for the y-axis.

56. Reasoning  The x-intercept of a line is 2 and the y-intercept is 4. Use this
information to write an equation for the line.

57. Coordinate Geometry  The vertices of a triangle are A(0, 0), B(2, 5), and C(4, 0).
a. Write an equation for the line through A and B.
b. Write an equation for the line through B and C.
c. Compare the slopes and the y-intercepts of the two lines.

Lesson 3-7  Equations of Lines in the Coordinate Plane 195

C Challenge Do the three points lie on one line? Justify your answer.

58. (5, 6), (3, 2), (6, 8) 59. ( -2, -2), (4, -4), (0, 0) 60. (5, -4), (2, 3), ( -1, 10)

Find the value of a such that the graph of the equation has the given slope.

61. y = 92ax + 6; m = 2 62. y = - 3ax - 4; m = 21 63. y = - 4ax - 10; m = - 2
3

Standardized Test Prep

SAT/ACT 64. AB has endpoints A(8, k) and B(7, -3). The slope of AB is 5. What is k? 8
1 2 5

65. Two angles of a triangle measure 68 and 54. What is the measure of the third angle?

14 58 122 180

66. Which of the following CANNOT be true? BC
plane ABCD } plane EFGH AD
Planes ABCD and CDHG intersect in <CD>.
FG
ABCD and ABC represent the same plane.
EH
Short plane ADHE } plane DCG
Response
67. The length of a rectangle is (x - 2) inches and the width is 5x inches. Which

expression represents the perimeter of the rectangle in inches?

6x - 2 12x - 4 5x2 - 1h0sxm 11gmse_03 0107x_2t0-02404x6.ai

68. One of the angles in a certain linear pair is acute. Your friend says the other angle
must be obtuse. Is your friend’s conjecture reasonable? Explain.

Mixed Review

For Exercises 69 and 70, construct the geometric figure. See Lesson 3-6.

69. a rectangle with a length twice its width 70. a square

Name the property that justifies each statement. See Lesson 2-5.
71. 4(2a - 3) = 8a - 12 72. If b + c = 7 and b = 2, then 2 + c = 7.
73. RS ≅ RS 74. If ∠1 ≅ ∠4, then ∠4 ≅ ∠1.

Get Ready!  To prepare for Lesson 3-8, do Exercises 75–77.

Find the slope of the line passing through the given points. See Lesson 3-7.

75. (2, 5), ( -2, 3) 76. (0, -5), (2, 0) 77. (1, 1), (2, -4)

196 Chapter 3  Parallel and Perpendicular Lines

3-8 Slopes of Parallel and CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Perpendicular Lines
MG-AGFPSE..9B1.25. GP-rGovPeEt.h2e.5s loPproevceritheeriaslfooprepcarriatellreial afnodr
paerpaellneldaicnudlaprelrinpesndaincduluasrelintheesmantdo usoselvethgeemomtoestroilcve
gpreoobmleemtrsic. problems.

MP 1, MP 3, MP 4

Objective To relate slope to parallel and perpendicular lines

Think about what You and a friend enjoy exercising together. Distance (mi) 2.0
the slope of a One day, you are about to go running when 1.5
line means in the your friend receives a phone call. You 1.0
context of this decide to start running and tell your friend 0.5
situation. to catch up after the call. The red line
represents you and the blue line represents 0
your friend. Will your friend catch up? 0 4 8 12 16
Explain. Time (min)

MATHEMATICAL

PRACTICES In the Solve It, slope represents the running rate, or speed. According toHtShMe1g1rGaMphS,Ey_0o3u08_a00846
and your friend run at the same speed, so the slopes of the lines are the 2snadmpea.sIsn1t2h-1is0-08
lesson, you will learn how to use slopes to determine how two lines relaDteurgkreaphically
to each other.

Essential Understanding  You can determine whether two lines are parallel or
perpendicular by comparing their slopes.

When two lines are parallel, their slopes are the same.

Key Concept  Slopes of Parallel Lines

• If two nonvertical lines are parallel, then their slopes are equal.
• If the slopes of two distinct nonvertical lines are equal, then the lines are parallel.
• Any two vertical lines or horizontal lines are parallel.

Lesson 3-8  Slopes of Parallel and Perpendicular Lines 197

Problem 1 Checking for Parallel Lines ᐍ2 ᐍ1 y
Are lines O1 and O2 parallel? Explain. (؊3, 3) (؊1, 5)
Can you tell from the Step 1 Find the slope of each line.
diagram whether the
lines are parallel? slope of /1 = 5 - ( - 4) = 9 = -3 1 x
No. The lines may look -1 - 2 -3 Ϫ4 O 24
parallel, but you only see
a small portion of their Step 2 slope of /2 = 3 - ( - 4) = 7 = - 7 Ϫ2 (2, ؊4)
graphs. Compare their - 3 - ( - 1) -2 2
slopes to know for sure. (؊1, ؊4)

Compare the slopes.

Since - 3 ≠ - 72, /1 and /2 are not parallel.

Got It? 1. Line /3 contains A( -13, 6) and B( -1, 2). Line /4 contains C(3, 6) and
D(6, 7). Are /3 and /4 parallel? Explain.
hsm11gmse_0308_t00292.ai

How does the given Problem 2 Writing Equations of Parallel Lines
line help you? What is an equation of the line parallel to y ∙ ∙3x ∙ 5 that contains (∙1, 8)?
Parallel lines have the
same slope. Once you Identify the slope of the given line. y ∙ ∙3x∙5
know the slope of the y ∙ y1 ∙ m(x ∙ x1)
given line, you know the You now know the slope of the y ∙ 8 ∙ ∙3(x ∙ (∙1))
slope you need to write new line and that it passes through y ∙ 8 ∙ ∙3(x ∙ 1)
an equation. (- 1, 8). Use point-slope form to
write the equation.

Substitute - 3 for m and (-1, 8)
for (x1, y1) and simplify.

Got It? 2. What is an equation of the line parallel to y = -x - 7 that
contains ( -5, 3)?

When two lines are perpendicular, the product of their slopes is -1. Numbers with product -1
are opposite reciprocals. This proof will be presented in more detail in Chapter 7.

Key Concept  Slopes of Perpendicular Lines
• If two nonvertical lines are perpendicular, then the product of their slopes is -1.
• If the slopes of two lines have a product of -1, then the lines are perpendicular.
• Any horizontal line and vertical line are perpendicular.

198 Chapter 3  Parallel and Perpendicular Lines

Problem 3 Checking for Perpendicular Lines

Can you tell from Lines O1 and O2 are neither horizontal nor vertical. Are ᐍ1 y ᐍ2
the diagram whether they perpendicular? Explain. 4 x
the lines are (4, 3)
perpendicular? Step 1 Find the slope of each line. (؊4, 2) 2

No. You can tell that m1 = slope of /1 = 2 - ( - 4) = 6 = - 3
-4 - 0 -4 2
the lines intersect, but
m2 = slope of /2 = 3 - ( - 3) = 6 = 2 Ϫ4 O 2 4
not necessarily at right 4 - ( - 5) 9 3 ᎐2

angles. So, you need to # #Step 2 Find the product of the slopes. (؊5, ؊3) (0, ؊4)

compare their slopes.

m1 m2 = - 3 2 = -1
2 3

Lines /1 and /2 are perpendicular because the product of
their slopes is -1.

Got It? 3. Line /3 contains A(2, 7) and B(3, -1). Line /4 contaihnssmC(1-12g, m6)saen_d0D3(088, 7_)t.00327.ai
Are /3 and /4 perpendicular? Explain.

How is this similar to Problem 4 Writing Equations of Perpendicular Lines
writing equations of What is an equation of the line perpendicular to y ∙ 15x ∙ 2 that contains (15, ∙4)?
parallel lines? Step 1 Identify the slope of the given line.
You follow the same
process. The only y = 51x + 2
difference here is that the       c
slopes of perpendicular slope
lines have product −1.

#Step 2 Find the slope of the line perpendicular to the given line.
m1 m2 = - 1 The product of the slopes of # lines is - 1.
#
1 m2 = - 1 Substitute 1 for m1.
5 5

m2 = - 5 Multiply each side by 5.

Step 3 Use point-slope form to write an equation of the new line.

y - y1 = m(x - x1)
y - ( - 4) = - 5(x - 15) Substitute - 5 for m and (15, - 4) for (x1, y1).
y + 4 = - 5(x - 15) Simplify.

Got It? 4. What is an equation of the line perpendicular to y = -3x - 5 that
contains ( -3, 7)?

Lesson 3-8  Slopes of Parallel and Perpendicular Lines 199

Problem 5 Writing Equations of Lines

Sports  The baseball field below is on a coordinate grid with home plate at the origin.
A batter hits a ground ball along the line shown. The player at (110, 70) runs along
a path perpendicular to the path of the baseball. What is an equation of the line on
which the player runs?

120 y
100
80
60
40
20

0x
0 20 40 60 80 100 120

Step 1 Find the slope of the baseball’s path.

m1 = y2 - y1 = 20 - 10 = 10 = 1 Points (30, 10) and (60, 20) are
Step 2 x2 - x1 60 - 30 30 3 on the baseball’s path.

Find the slope of a line perpendicular to the baseball’s path.
#m1 m2 = - 1 The product of the slopes of # lines is - 1.
1
m2 = - 1 Substitute 1 for m1.
#3 3

Which linear m2 = - 3 Multiply each side by 3.
equation form
should you use? Step 3 Write an equation of the line on which the player runs.
You know the slope. The slope is -3 and a point on the line is (110, 70).
The player is located
at a point on the line. y - y1 = m(x - x1) Point-slope form
Use point-slope form.
y - 70 = - 3(x - 110) Substitute - 3 for m and (110, 70) for (x1, y1).

Got It? 5. Suppose a second player standing at (90, 40) misses the ball, turns around,
and runs on a path parallel to the baseball’s path. What is an equation of the
line representing this player’s path?

200 Chapter 3  Parallel and Perpendicular Lines

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
<AB> contains points A and B. <CD> contains points C and
D. Are <AB> and <CD> parallel, perpendicular, or neither? 5. Error Analysis  Your classmate tries to find an
Explain. equation for a line parallel to y = 3x - 5 that
1. A( -8, 3), B( -4, 11), C( -1, 3), D(1, 2) contains ( -4, 2). What is your classmate’s error?
2. A(3, 5), B(2, -1), C(7, -2), D(10, 16)
3. A(3, 1), B(4, 1), C(5, 9), D(2, 6) slope of given line ‫ ؍‬3

4. What is an equation of the line perpendicular to slope of parallel line ‫؍‬ 1
y = -4x + 1 that contains (2, -3)? 3
y ؊ y1 ‫ ؍‬m(x ؊ x1)
1
y ؊ 2 ‫؍‬ 3 (x ؉ 4)

6. Compare and Contrast  What are the differences
between the equations of parallel lines and the

equationhssomf p1e1rgpmensdeic_u0la3r0l8in_ets0?0E4x5pl1a.iani.

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice For Exercises 7–10, are lines O1 and O2 parallel? Explain. See Problem 1.

7. y 8. 2 y (1, 3) ᐍ1(4, 2)ᐍ2
(؊5, 1) x
(؊6, 1) (0, 3)
Ϫ6 Ϫ4 Ϫ1 O 2 4 6
1 (4, 1) x
24 ᐍ2
Ϫ6 O

(0, ؊2) ᐍ1 (؊4, ؊2)

9. y 10. ᐍ1 y ᐍ2
3 (5, 3) 4
(؊2, 5)
hsm11g(m0,s2e)_0308_t00454.ai
(h؊s3m, 11)1gmse2_030(18,_2t)00455.ai
x
x
Ϫ4 O 2 4
Ϫ4 Ϫ2 O 2 4
Ϫ2
Ϫ2 (0, ؊2)
ᐍ1 (؊4, ؊4) (2, ؊3)
ᐍ2

Write an equation of the line parallel to the given line that contains C. See Problem 2.

11. Ch(0s,m3)1;1yg=m-s2ex_+0310 8_t00456.ai 12. Ch(6s,m0)1; 1y g=m13xse_0308_t00457.ai

13. C( -2, 4); y = 12x + 2 14. C(6, -2); y = - 32x + 6

Lesson 3-8  Slopes of Parallel and Perpendicular Lines 201

For Exercises 15–18, are lines O1 and O2 perpendicular? Explain. See Problem 3.
x
15. y ᐍ2 16. 4 y (2, 4)
ᐍ1 1 (4, 1) x (؊3, 4)
Ϫ1 O 2 4
(؊4, 0) O
(0, ؊2) (4, ؊4)
Ϫ4 Ϫ2 2 4
ഞ2 Ϫ2 (1, ؊2)

(1, ؊5) Ϫ4 ഞ1

17. (؊4, 4) y (6, 3) 18. 4y (3, 4)
(؊2, 4)

hsm11gm2se_0308_t00461.ai hsm11gmse_0308_t00463.ai

x (2, 0) x

Ϫ4 Ϫ2 O (1, ؊1) 5 Ϫ4 O 4

ᐍ2 Ϫ4 (3, ؊3) ᐍ1 (؊3, ؊2) ᐍ1
ᐍ2 Ϫ4

Write an equation of the line perpendicular to the given line that contains P. See Problem 4.

19. Ph(6s,m6)1; 1yg=m23xs e_0308_t004662.a0i. P(4, 0); y = 21x - 5 hsm11gms2e1_.0 P3(048, 4_)t;0y0=46-92.xai- 8

STEM 22. City Planning  City planners want to 3 y See Problem 5.
construct a bike path perpendicular
Bruckner Blvd.
to Bruckner Boulevard at point P. An x

equation of the Bruckner Boulevard line Ϫ4 Ϫ2 P 2 4
is y = -34x. Find an equation of the line Ϫ2
for the bike path.

B Apply Rewrite each equation in slope-intercept
form, if necessary. Then determine whether the lines are parallel. Explain.

23. y = -x + 6 24. y - 7x = 6 hsm251 . 136gxxm++s42eyy_==031620 8_t004702.6a .i 12x0y+=5y- = -1
4x - 20
x + y = 20 y + 7x = 8

27. Think About a Plan  Line /1 contains ( -4, 1) and (2, 5) and line /2 contains (3, 0)
and ( -3, k). What value of k makes /1 and /2 parallel?

• For /1 and /2 to be parallel, what must be true of their slopes?
• What expressions represent the slopes of /1 and /2?
28. Open-Ended  Write equations for two perpendicular lines that have the same

y-intercept and do not pass through the origin.

29. Writing  Can the y-intercepts of two nonvertical parallel lines be the same? Explain.

202 Chapter 3  Parallel and Perpendicular Lines

Use slopes to determine whether the opposite sides of quadrilateral ABCD
are parallel.

30. A(0, 2), B(3, 4), C(2, 7), D( -1, 5) 31. A( -3, 1), B(1, -2), C(0, -3), D( -4, 0)

32. A(1, 1), B(5, 3), C(7, 1), D(3, 0) 33. A(1, 0), B(4, 0), C(3, -3), D( -1, -3)

34. Reasoning  Are opposite sides of hexagon RSTUVW at the right parallel? y

Justify your answer. 4 RS

35. Which line is perpendicular to 3y + 2x = 12? 2x + 3y = 6 WT
6x - 4y = 24 y = -2x + 6
y + 3x = -2 x
O 1V U 6

Rewrite each equation in slope-intercept form, if necessary. Then determine
whether the lines are perpendicular. Explain.

36. y = -x - 7 37. y = 3 38. 2x -hs7my =11-g4m2 se_0308_t00471.ai
y - x = 20 x = -2
4y = -7x - 2

Developing Proof  Explain why each theorem is true for three lines in the
coordinate plane.

39. Theorem 3-7: If two lines are parallel to the same line, then they are parallel to
each other.

40. Theorem 3-8: In a plane, if two lines are perpendicular to the same line, then they
are parallel to each other.

41. Rail Trail  A community recently converted an y
old railroad corridor into a recreational trail. The
150

graph at the right shows a map of the trail on a 125 Recreational Trail
coordinate grid. They plan to construct a path to Parking lot
connect the trail to a parking lot. The new path will
be perpendicular to the recreational trail. 100

a. Write an equation of the line representing the 75

new path. 50
b. What are the coordinates of the point at which

the path will meet the recreational trail? 25
c. If each grid space is 25 yd by 25 yd, how long is
x
the path to the nearest yard?
OO 25 50 75 100 125 150

42. Reasoning  Is a triangle with vertices G(3, 2),

H(8, 5), and K(0, 10) a right triangle? Justify your answer. f1in).d<CtDh>e4HcstoSlhMonppt1aae1isnGos1sMf 2<AS-0BE6_>.-003808_a01164
43. Graphing Calculator  <AB> contains points A( -3, 2) and B(5,
points C(2, 7) and D(1, -1). Use your graphing calculator to

Enter the x-coordinates of A and B into the L1 list of your list editor. DEnurtkeer the

y-coordinates into the L2 list. In your stat fCinAdLtChemselonpuesoefle<CcDt L>.iAnrRee<AgB(>aaxn+d b<C)D. >
Press enter to find the slope a. Repeat to

parallel, perpendicular, or neither?

Lesson 3-8  Slopes of Parallel and Perpendicular Lines 203

C Challenge For Exercises 44 and 45, use the graph at the right. y A
10 B
44. Show that the diagonals of the figure are congruent.

45. Show that the diagonals of the figure are perpendicular
bisectors of each other.

46. a. Graph the points P(2, 2), Q(7, 4), and R(3, 5). 5 D C x
b. Find the coordinates of a point S that, along with points O 5 10

P, Q, and R, will form the vertices of a quadrilateral with
opposite sides parallel. Graph the quadrilateral.
c. Repeat part (b) to find a different point S. Graph the
new quadrilateral.

47. Algebra  A triangle has vertices L( -5, 6), M( -2, -3), and N(4, 5). Write an
equation for the line perpendicular to LM that contains point N.

hsm11gmse_0308_t00474.ai

Standardized Test Prep

SAT/ACT 48. △ABC is right with right angle C. The slope of AC is -2. What is the slope of BC?

49. In the diagram at the right, M is the midpoint of AB. What is AB? x2 Ϫ 6 x

50. What is the distance between ( -4.5, 1.2) and (3.5, -2.8) to the AM B
nearest tenth?
(x2 Ϫ 14)Њ
51. What is the value of x in the diagram at the right?
hsm11gmse_0308_t00476.ai
52. The perimeter of a square is 20 ft. What is the area of the square
in square feet? 5xЊ

Mixed Review

Algebra  Write an equation for the line containing the given points. hsm11gmse_03S0e8e_Lte0s0s4o7n73.a-7i.

53. A(0, 3), B(6, 0) 54. C( -4, 2), D( -1, 7) 55. E(3, -2), F( -5, -8)

Name the property that justifies each statement. See Lesson 2-5.
56. ∠4 ≅ ∠4 57. If m∠B = 8, then 2m∠B = 16.
58. -3x + 6 = 3( -x + 2) 59. If RS ≅ MN , then MN ≅ RS.

G et Ready!  To prepare for Lesson 4-1, do Exercises 60–62.

Are j1 and j2 congruent? Explain. See Lessons 1-5 and 2-6.
54Њ
60. 1 61. 62.
2 12 12

204 Chapter 3  Parallhelsamnd11Pgermpesned_ic0u3la0r8L_int0es0497.ai hsm11gmse_0308_t00498.ai hsm11gmse_0308_t00499.ai

3 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 3-1,
problems, you 3-2, and 3-5. Use the work you did to complete the following.
will pull together 1. Solve the problem in the Task Description on page 139 by finding the measure of each
many concepts
and skills that numbered angle in the blueprint. Show all your work and explain each step of your
you have studied solution.
about parallel
lines. 2. Reflect  Choose one of the Mathematical Practices below and explain how you applied
it in your work on the Performance Task.

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.

MP 7: Look for and make use of structure.

On Your Own

A subcommittee of the city’s planning department suggests a different plan for the walkways
in the city’s new park. In the revised blueprint shown below, the walkways are shown in blue
and parallel lines are indicated by red arrowheads. The right angles of the rectangular park
are marked, and two other angle measures are given in red.

2 35
14

67
89

10 11 12 13
14 15 16 132°

17 18

42° 22
19 20 21

Find the measure of each numbered angle in the blueprint.

Chapter 3  Pull It All Together 205

3 Chapter Review

Connecting and Answering the Essential Questions

1 Reasoning and Parallel Lines and Angle Pairs

Proof (Lessons 3-2 and 3-3) Parallel and
You can prove that two Perpendicular Lines
lines are parallel by using ∠1 ≅ ∠5 1 2 (Lesson 3-4)
special angle relationships ∠4 ≅ ∠6 4 a } b and b } c S a } c
and the relationships of two m∠4 + m∠5 = 180 56 a # b and a # c S b } c
lines to a third line. ∠2 ≅ ∠8

8

2 Measurement Parallel Lines and Triangles (Lesson 3-5)
The sum of the measures B
of the angles of a triangle
is 180.   m∠A +hmsm∠B11+gmm∠seC_=0318c0r_t00514.ai

3 Coordinate AC
Geometry
You can write the equation Lines in the Coordinate Plane (Lesson 3-7) Slopes of Parallel and
of a line by using its slope Perpendicular Lines
and a point on the line. hSPlosoimpnet-1-sinl1otgperemcefospretmf_o:0r m3:c  ryy_=-t0my012x=6+5mb(x - x1) (Lesson 3-8)

Parallel lines: equal slopes
Perpendicular lines:
product of slopes is -1

Chapter Vocabulary • flow proof (p. 158) • same-side interior angles (p. 142)
• parallel lines (p. 140) • skew lines (p. 140)
• alternate exterior angles (p. 142) • parallel planes (p. 140) • slope (p. 189)
• alternate interior angles (p. 142) • point-slope form (p. 190) • slope-intercept form (p. 190)
• auxiliary line (p. 172) • remote interior angles (p. 173) • transversal (p. 141)
• corresponding angles (p. 142)
• exterior angle of a polygon (p. 173)

Choose the correct term to complete each sentence.

1. A(n) ? intersects two or more coplanar lines at distinct points.

2. The measure of a(n) ? of a triangle is equal to the sum of the measures of its two
remote interior angles.

3. The linear equation y - 3 = 4(x + 5) is in ? .
4. When two coplanar lines are cut by a transversal, the angles formed between the

two lines and on opposite sides of the transversal are ? .

5. Noncoplanar lines that do not intersect are ? .

6. The linear equation y = 3x - 5 is in ? .

206 Chapter 3  Chapter Review

3-1  Lines and Angles

Quick Review Exercises

A transversal is a line that intersects two or more coplanar Identify all numbered angle pairs that form the given type

lines at distinct points. of angle pair. Then name the two lines and transversal that

∠1 and ∠3 are corresponding form each pair.

angles. 18 7. alternate interior angles ab
27 8. same-side interior angles 5 81 c
∠2 and ∠6 are alternate interior 9. corresponding angles 3
angles. 36 10. alternate exterior angles
45 26 4 d
∠2 and ∠3 are same-side interior Classify the angle pair formed by 7e
angles.

∠4 and ∠8 are alternate exterior angles.

Example j1 and j2.

Name two other pairs of correspondinghasnmg1le1sginmthsee_03cr_t005 111.5 .ai 12.

diagram above. 2 hs1m11gmse_03cr_t00516
∠5 and ∠7 1
2

∠2 and ∠4

3-2  Properties of Parallel Lines hsm11gmse_03cr_t00517h.asmi 11gmse_03cr_t00518.ai

Quick Review Exercises

If two parallel lines are cut by a transversal, then Find mj1 and mj2. Justify your answers.

• corresponding angles, alternate interior angles, and 13. 2 a 14. r
alternate exterior angles are congruent 1 q
21
• same-side interior angles are supplementary 105Њ

bp

Example c 120Њ

Which other angles measure 110? 2 6 7 15. Find the values of x and y in the diagram below.
∠6 (corresponding angles) 3
∠3 (alternate interior angles) hsm11gmse_03cr_ytЊ00520h.asmi 11gmse_03cr_t00521.ai
∠8 (vertical angles) 51108Њ4
118Њ xЊ
25Њ

hsm11gmse_03cr_t00519.ai hsm11gmse_03cr_t00522.ai

Chapter 3  Chapter Review 207

3-3  Proving Lines Parallel

Quick Review Exercises

If two lines and a transversal form Find the value of x for which O } m.
• congruent corresponding angles,
16. ᐍ m 17.
• congruent alternate interior angles, ᐍ
65Њ 130Њ
• congruent alternate exterior angles, or (3x ϩ 5)Њ m (2x ϩ 10)Њ

• supplementary same-side interior angles,

then the two lines are parallel.

Example Use the given information to decide which lines, if any,
are parallel. Justify your conclusion.
What is the value of x for which ᐍ
O } m? 2x Њ m 18. ∠hs1m≅1∠1g9mse_03cr_t00524h.asim11ᐍgmse_03mcr_t00525.ai

The given angles are alternate 106Њ 19. m∠3 + m∠6 = 180 12 3 4
interior angles. So, / } m if the
given angles are congruent. 20. m∠2 + m∠3 = 180 87 65 n
21. ∠5 ≅ ∠11 9 10
11 12
2x = 106 Congruent ⦞ have equal measures. p

x = 53 Divide each side by 2.

hsm11gmse_03cr_t00523.ai

3-4  Parallel and Perpendicular Lines hsm11gmse_03cr_t00526.ai

Quick Review Exercises ab c
d
• Two lines } to the same line are } to each other. Use the diagram at the right
• In a plane, two lines # to the same line are }. to complete each statement.
• I n a plane, if one line is # to one of two } lines,
22. If b # c and b # d, then
then it is # to both } lines. c ? d.

Example 23. If c } d, then ? # c.

What are the pairs of parallel and perpendicular lines in 24. Maps  Morris Avenue intersects both 1st Street

the diagram? a and 3rd Street at right angles. 3rd Street is parallel

/ } n, / } m, and m } n. to 5th Street. How are 1st Streethasnmd151thgSmtrseeet_03cr_t00528.ai
a # /, a # m, and a # n. related? Explain.
ᐍ
m

n

hsm11gmse_03cr_t00527.ai

208 Chapter 3  Chapter Review

3-5  Parallel Lines and Triangles

Quick Review Exercises

The sum of the measures of the angles of a triangle is 180. Find the values of the variables.

The measure of each exterior angle of a triangle equals the 25. 26.
sum of the measures of its two remote interior angles. 60Њ xЊ

Example xЊ yЊ 120Њ yЊ 135Њ

What are the values of x and y?

x + 50 = 125 Exterior Angle 50Њ The measures of the three angles of a triangle are given.
Theorem 125Њ yЊ xЊ
F 2i7n.d xht,hs2mex,v13ax1lugemosf ex._03cr_t00530h.asmi 11gmse_03cr_t00531.ai
x = 75  Simplify.

x + y + 50 = 180 Triangle Angle-Sum Theorem 28. x + 10, x - 20, x + 25

75 + y + 50 = 180 Substitute 75 for x. hsm11gmse_03cr_t0 2095. 2290x.a+i 10, 30x - 2, 7x + 1

y = 55 Simplify.

3-6  Constructing Parallel and Perpendicular Lines

Quick Review Exercises

You can use a compass and a straightedge to construct 3 0. D raw a line m and point Q not on m. Construct a line
perpendicular to m through Q.
• a line parallel to a given line through a point not on
the line Use the segments below.
a
• a line perpendicular to a given line through a point on the
line, or through a point not on the line b

Example 31. C onstruct a rectangle with side lengths a and b.
32. C onstruct a rectangle with side lengths a and 2b.
Which step of the parallel lines construction guarantees
the lines are parallel? 33. C hosnmst1ru1cgtma qseu_ad0r3ilcart_erta0l0w5i3th2o.anie pair of parallel

The parallel lines construction involves G opposite sides, each side of length 2a.
constructing a pair of congruent Cᐍ
angles. Since the congruent angles are
corresponding angles, the lines are
parallel.

hsm11gmse_0306_t00495.ai

Chapter 3  Chapter Review 209

3-7  Equations of Lines in the Coordinate Plane

Quick Review Exercises

Slope-intercept form is y = mx + b, where m is the slope Find the slope of the line passing through the points.
and b is the y-intercept.
Point-slope form is y - y1 = m(x - x1), where m is the 34. (6, - 2), (1, 3) 35. ( - 7, 2), ( - 7, - 5)
slope and (x1, y1) is a point on the line.
36. Name the slope and y-intercept of y = 2x - 1.
Example Then graph the line.

What is an equation of the line with slope −5 and 37. Name the slope of and a point on y - 3 = - 2(x + 5).
y-intercept 6? Then graph the line.
Use slope-intercept form:  y = -5x + 6.
Write an equation of the line.
Example 38. slope - 21, y-intercept 12
39. slope 3, passes through (1, - 9)
What is an equation of the line through (−2, 8) with
slope 3? 40. passes through (4, 2) and (3, - 2)
Use point-slope form:  y - 8 = 3(x + 2).

3-8  Slopes of Parallel and Perpendicular Lines

Quick Review Exercises

Parallel lines have the same slopes. Determine whether <AB> and <CD> are parallel,
The product of the slopes of two perpendicular lines is -1. perpendicular, or neither.

Example 41. A( -1, -4), B(2, 11), C (1, 1), D(4, 10)
42. A(2, 8), B( -1, -2), C (3, 7), D(0, -3)
What is an equation of the line perpendicular to 43. A( -3, 3), B(0, 2), C (1, 3), D( -2, -6)
y = 2x − 5 that contains (1, −3)? 44. A( -1, 3), B(4, 8), C ( -6, 0), D(2, 8)

Step 1 Identify the slope of y = 2x - 5.

The slope of the given line is 2. 45. W rite an equation of the line parallel to y = 8x - 1
that contains ( -6, 2).
Step 2 Find the slope of a line perpendicular to

y = 2x - 5. 46. W rite an equation of the line perpendicular to

( )

Step 3
The slope is - 21, because 2 - 1 = -1. y = 1 x + 4 that contains (3, - 3).
U se point-slope 2 +3=- 6
21(x -
form to write y 1).

210 Chapter 3  Chapter Review