Geometry

Get Ready! CHAPTER

Lesson 1-7 The Distance Formula 4

Find the side lengths of △ABC.

1. A(3, 1), B( -1, 1), C( -1, -2)

2. A( -3, 2), B( -3, -6), C(8, 6)

3. A( -1, -2), B(6, 1), C(2, 5)

Lesson 2-6 Proving Angles Congruent

Draw a conclusion based on the information given.

4. ∠J is supplementary to ∠K ; 5. ∠M is supplementary to ∠N ;
∠L is supplementary to ∠K . ∠M ≅ ∠N.

6. ∠1 is complementary to ∠2. 7. FA> # FC>, FB> # FD>

A 1 B C

B C AD
2 F

Lessons 3-2 Parallel Lines and the Triangle Angle-Sum Theorem

and 3-5 What can you conclude about the angles in each diagram?

8. hsm11gmse_04cCo_t092.5 69 A hsm11Bgm10s.e _04(xcϩo_9t)Њ02570 C (7x ϩ 4)Њ

A C A (6x Ϫ 1)Њ
E DD B

B

hsm11gmse_04co_t02573

Looking Ahead Vocahbsmul1a1grmyse_04co_t02572

11. Thhsemfo1u1ngdmatisoen_o0f4acbou_iltd0i2n5g 7is1the base of the building. How would you describe

the base of an isosceles triangle in geometry?

12. The legs of a table support the tabletop and are equal in length. How might they be
similar to the legs of an isosceles triangle?

13. A postal worker delivers each piece of mail to the mailbox that corresponds to the
address on the envelope. What might the term corresponding parts of geometric
figures mean?

Chapter 4  Congruent Triangles 215

4CHAPTER Congruent Triangles

Download videos VIDEO Chapter Preview 1 Visualization
connecting math Essential Question  How do you identify
to your world.. 4-1 Congruent Figures corresponding parts of congruent triangles?
4-2 Triangle Congruence by SSS and SAS
Interactive! ICYNAM 4-3 Triangle Congruence by ASA and AAS 2 Reasoning and Proof
Vary numbers, ACT I V I TI 4-4 Using Corresponding Parts of Congruent Essential Question  How do you show
graphs, and figures D that two triangles are congruent?
to explore math ES Triangles
concepts.. 4-5 Isosceles and Equilateral Triangles 3 Reasoning and Proof
4-6 Congruence in Right Triangles Essential Question  How can you
4-7 Congruence in Overlapping Triangles tell whether a triangle is isosceles or
equilateral?
The online
Solve It will get
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Congruence
Spanish English/Spanish Vocabulary Audio Online:
• Mathematical Practice: Construct viable
English Spanish arguments

Online access base angles of an ángulos de base de un • Modeling with Geometry
to stepped-out isosceles triangle, p. 250 triángulo isósceles
problems aligned
to Common Core base of an isosceles base de un triángulo
Get and view triangle, p. 250 isósceles
your assignments
online. congruent polygons, p. 219 polígonos congruentes

NLINE corollary, p. 252 corolario
ME WO
O hypotenuse, p. 258 hipotenusa
RK
HO legs of an isosceles catetos de un
triangle, p. 250 triángulo isósceles

Extra practice legs of a right catetos de un
and review triangle, p. 258 triángulo rectángulo
online
vertex angle of an ángulo en vértice de
isosceles triangle, p. 250 un triángulo isósceles

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Estimating the Distance Across a Gorge

Jamal wants to estimate the distance across the gorge shown in the diagram
below. He stands at point Y and locates a tree at point X directly across the gorge
to the north. He then walks west along the gorge 500 ft and marks point A. After
walking another 500 ft in the same direction, he turns 90° and walks south,
perpendicular to the gorge. He stops when his location appears to form a straight
line with points A and X. Jamal measures the distance BC as 327 ft.

X

N
B A Y WE

S

C

Task Description

Estimate the distance XY across the gorge.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll label a diagram to help you identify facts about parts of the diagram. (MP 1)

• You’ll construct a viable argument to prove facts about the diagram. (MP 3)

• You’ll look for and use relationships in the diagram. (MP 7)

Chapter 4  Congruent Triangles 217

4-1 Congruent Figures MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Prepares for GM-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce . . . criteria
cfoorntgrriaunegnlcees.t.o.scoriltveeriparofobrletrmiasnagnledsptroosvoelvrelaptrioobnlsehmipss in
agneodmperotrviec frieglautrieosn. ships in geometric figures.

MP 1, MP 3, MP 4, MP 7

Objective To recognize congruent figures and their corresponding parts

You are working on a puzzle. You’ve almost finished, except for a few
pieces of the sky. Place the remaining pieces in the puzzle. How did you
figure out where to place the pieces?

Having trouble? A 1
How can tracing C
pieces 1, 2, and 3
help? B
2
MATHEMATICAL

PRACTICES

3

Lesson Congruent figures have the same size and shape. When two figures are congruent, you
can slide, flip, or turn one so that it fits exactly on the othHeSrMo1n1eG, aMsSsEh_o0w4n01b_ealo0w22. 8In0 this
Vocabulary lesson, you will learn how to determine if geometric figu2rneds apraescso1n1g-1ru7e-0n8t.
• congruent
Durke
polygons Slide Turn

Flip

Essential Understanding  You can determine whether two figures are congruent
by comparing their corresponding parts.

hsm11gmse_0401_t02403

218 Chapter 4  Congruent Triangles

Key Concept  Congruent Figures

Definition Example E AB ≅ EF     BC ≅ FG
Congruent polygons have A BF H CD ≅ GH    DA ≅ HE
congruent corresponding
parts—their matching sides D CG ∠A ≅ ∠E     ∠B ≅ ∠F
and angles. When you name ABCD ≅ EFGH ∠C ≅ ∠G     ∠D ≅ ∠H
congruent polygons, you
must list corresponding
vertices in the same order.

hsm11gmse_0401_t02405 L
M
Problem 1 Finding Congruent Parts K JO
N
How do you know If HIJK ≅ LMNO, what are the congruent corresponding parts? H
which sides and Sides: HI ≅ LM IJ ≅ MN JK ≅ NO KH ≅ OL I
angles correspond?

The congruence statement Angles: ∠H ≅ ∠L ∠I ≅ ∠M ∠J ≅ ∠N ∠K ≅ ∠O
HIJK ≅ LMNO tells you

which parts correspond.
Got It? 1. If △WYS ≅ △MKV, what are the congruent corresponding parts?

hsm11gmse_0401_t02407

You know two angle Problem 2 Using Congruent Parts
measures in △ABC. Multiple Choice  The wings of an SR-71 Blackbird aircraft suggest
How can they help? congruent triangles. What is mjD?

In t he congruent 30 75 105 150
triangles, ∠D
corresponds to ∠A, Use the Triangle Angle- m∠A + 30 + 75 = 180
so you know that Sum Theorem to write an
∠D ≅ ∠A. You can find equation involving
m∠D by first finding m∠A.
m∠A.

Solve for m∠A. m∠A + 105 = 180
m∠A = 75
∠A and ∠D are
corresponding parts of m∠A = m∠D = 75
congruent triangles, so The correct answer
∠A ≅ ∠D. is B.

Got It? 2. Suppose that △WYS ≅ △MKV. If m∠W = 62 and m∠Y = 35, what is
m∠V ? Explain.

Lesson 4-1  Congruent Figures 219

Problem 3 Finding Congruent Triangles

How do you Are the triangles congruent? Justify your answer. B4 6 E
determine whether
two triangles are AB ≅ ED Given
congruent?
Compare each pair of BC ≅ DC BC = 4 = DC 6 C4 D
corresponding parts. If all
six pairs are congruent, AC ≅ EC AC = 6 = EC A
then the triangles are
congruent. ∠A ≅ ∠E, ∠B ≅ ∠D Given

∠BCA ≅ ∠DCE Vertical angles are congruent.

△ABC ≅ △EDC by the definition of congruent triangles. hsm11Dgmse_0401_t02409

Got It? 3. Is △ABD ≅ △CBD? Justify your answer.

ABC

Recall the Triangle nAenxgtleth-SeuomremThfeoollroewms:fTrohme stuhme Torfitahnegmle eAhanssgumlree1-sS1uogfmmthTesheae_no0grel4em0s.1in_at02414
triangle is 180. The

Theorem 4-1  Third Angles Theorem

Theorem If . . . Then . . .
If two angles of one triangle ∠A ≅ ∠D and ∠B ≅ ∠E ∠C ≅ ∠F
are congruent to two
angles of another triangle, AD
then the third angles are
congruent. B CE F

Proof Proof of Theorem 4-1:  Third Angles Theorem A D
F
Given:  ∠A ≅ ∠D, ∠B ≅ ∠E hsm11gmse_0401_t0B2416 C E
Prove:  ∠C ≅ ∠F

Statements Reasons

1) ∠A ≅ ∠D, ∠B ≅ ∠E hsm 121)) 1DGgievmfe. nosef _≅04⦞01_t05007
2) m∠A = m∠D, m∠B = m∠E
3) m∠A + m∠B + m∠C = 180, 3) △ Angle-Sum Thm.

m∠D + m∠E + m∠F = 180 4) Subst. Prop.
4) m∠A + m∠B + m∠C = m∠D + m∠E + m∠F 5) Subst. Prop.
5) m∠D + m∠E + m∠C = m∠D + m∠E + m∠F 6) Subtraction Prop. of =
6) m∠C = m∠F 7) Def. of ≅ ⦞
7) ∠C ≅ ∠F

220 Chapter 4  Congruent Triangles

Proof Problem 4 Proving Triangles Congruent M N
L
You know four pairs Given:  LM ≅ LO, MN ≅ ON,
of congruent parts. ∠M ≅ ∠O, ∠MLN ≅ ∠OLN O
What else do you
need to prove the Prove:  △LMN ≅ △LON
triangles congruent?
You need a third pair of Statements Reasons
congruent sides and a
third pair of congruent 1) LM ≅ LO, MN ≅ ON 1) Given
angles. 2) LN ≅ LN 2) Reflexive Property of ≅
3) ∠M ≅ ∠O, ∠MLN ≅ ∠OLN 3) Given
4) ∠MNL ≅ ∠ONL 4) Third Angles Theorem hsm11gmse_0401_t02421
5) △LMN ≅ △LON 5) Definition of ≅ triangles

Got It? 4. Given:  ∠A ≅ ∠D, AE ≅ DC, A C
B
EB ≅ CB, BA ≅ BD
Prove:  △AEB ≅ △DCB ED

Lesson Check hsm11gmse_0401_t02423

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

Complete the following statements. PRACTICES

1. Given:  △QXR ≅ △NYC 5. Open-Ended  When do you think you might need to
a. QX ≅ ?
b. ∠Y ≅ ? know that things are congruent in your everyday life?

2. Given:  △BAT ≅ △FOR 6. If each angle in one triangle is congruent to its
a. TA ≅ ? corresponding angle in another triangle, are the two
b. ∠R ≅ ? triangles congruent? Explain.

3. Given:  BAND ≅ LUCK 7. Error Analysis  Walter sketched the diagram
a. ∠U ≅ ? below. He claims it shows that the two polygons are
b. DB ≅ ? congruent. What information is missing to support
c. NDBA ≅ ? his claim?

4. In △MAP and △TIE, ∠A ≅ ∠I and ∠P ≅ ∠E.
a. What is the relationship between ∠M and ∠T ?
b. If m∠A = 52 and m∠P = 36, what is m∠T ?

hsm11gmse_0401_t02424 221

Lesson 4-1  Congruent Figures

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice 8. Construction  Builders use the king post truss (below left) for the top of a simple See Problem 1.
structure. In this truss, △ABC ≅ △ABD. List the congruent corresponding parts.

A

E H
G J

CBD FI

9. The attic frakminegtrpuossst(atbruosvse right) provides open attic frame truss In
space in the center for storage.

this truss, △EFG ≅ △HIJ. List the congruent corresponding parts.

△LMhCs≅m1△1BgJmK.sCeo_m04pl0e1te_tth0e2c4o2n7gruence statements. MJ
10. LC ≅ ? 11. KJ ≅ ? L CK

12. JB ≅ ? 13. ∠L ≅ ? B

14. ∠K ≅ ? 15. ∠M ≅ ?

16. △CML ≅ ? 17. △KBJ ≅ ?

18. △MLC ≅ ? 19. △JKB ≅ ? hsm11gmse_0401_t02429

POLY @ SIDE. List each of the following. 21. four pairs of congruent angles
20. four pairs of congruent sides

At an archeological site, the remains of two ancient step pyramids See Problem 2.
are congruent. If ABCD @ EFGH, find each of the following. (Diagrams are not
to scale.)

22. AD 23. GH B C F 280 ft G
24. m∠GHE 25. m∠BAD 45 ft 128Њ 45 ft 52Њ 128Њ
26. EF 27. BC A E
28. m∠DCB 29. m∠EFG 52Њ 335 ft H
D

For Exercises 30 and 31, can you conclude that the triangles are congruent? See Problem 3.

Justify your answers. HSM11GMSE_0401_a02282 HSM11GMSE_0401_a02283

30. △TRK and △TUK 2Dnudrkpeass3111.- 1△9-S0P8Q and △TUV 2nd pass 11-19-08
T Durke
S 7 V
R
U 5 P48

Q6 T7 U

K

222 Chapter 4  Congruhesnmt T1r1iagnmglesse_0401_t02432 hsm11gmse_0401_t02433

32. Given:  AB } DC, ∠B ≅ ∠D, B C See Problem 4.
Proof AB ≅ DC, BC ≅ AD
L
Prove:  △ABC ≅ △CDA E M
AD

B Apply 33. If △DEF ≅ △LMN, which of the following must be a correct
congruence statement?

DE ≅ LN ∠N ≅ ∠∠FhFsm11gmse_0401_t024D34 F
FE ≅ NL ∠M ≅ N

34. Reasoning  Randall says he can use the information in the figure C
to prove △BCD ≅ △DAB. Is he correct? Explain.
D
Algebra △ABC @ △DEF. Find the measures of the given angles or
the lengths of the given sides. Bhsm11gmse_0401_t02435

A

35. m∠A = x + 10, m∠D = 2x 36. m∠B = 3y, m∠E = 6y - 12

37. BC = 3z + 2, EF = z + 6 38. AC = 7a + 5, DF = 5a + 9

39. Think About a Plan  △ABC ≅ △DBE. Find the value of x. hsm11gmsCe_0401_t02436

• What does it mean for two triangles to be congruent? 51Њ E
• Which angle measures do you already know? B 81Њ
• How can you find the missing angle measure in a triangle?
A (x ϩ 5)Њ
Algebra Find the values of the variables.

40. C M 41. D
3xЊ A
D
45Њ
A 4 in. B L 2t in. K 6xЊ C
30Њ
᭝ABC Х ᭝KLM
B hsm11gmse_0401_t02437

᭝ACD Х ᭝ACB

42. Complete in two different ways: M Z

△hJsLmM1≅1gm? se. _0401_t02438 hsm11gLmse_0401J_t0N2439 R

43. Open-Ended  Write a congruence statement for
two triangles. List the congruent sides and angles.

44. Given:  AB # AD, BC # CD, AB ≅ CD, AD ≅ CB, AB } CD A B

Proo f Prove:  △ABD ≅ △CDB hsm11gmse_0401_t02440

DC

hsm11gmse_0401_t02442 223

Lesson 4-1  Congruent Figures

45. Given:  PR } TQ, PR ≅ TQ, PS ≅ QS, PQ bisects RT P R
Proo f Prove:  △PRS ≅ △QTS TS

46. Writing  The 225 cards in Tracy’s sports card collection are rectangles
of three different sizes. How could Tracy quickly sort the cards?

C Challenge Coordinate Geometry  The vertices of △GHJ are G(−2, −1), H(−2, 3), and J(1, 3). Q

47. △KLM ≅ △GHJ . Find KL, LM, and KM.

48. If L and M have coordinates L(3, -3) and M(6, -3), how many pairs of coordinates

are possible for K? Find one such pair. hsm11gmse_0401_t02441

49. a. How many quadrilaterals (convex and concave) with different shapes or sizes can

you make on a three-by-three geoboard? Sketch them. One is shown at the right.

b. How many quadrilaterals of each type are there?

PERFORMANCE TASK MATHEMATICAL hsm11gmse_0401_t0

Apply What You’ve Learned PRACTICES
MP 1
Look back at the information given on page 217 about how Jamal located the
points in the diagram. The diagram is shown again below.
X

N
B A Y WE

S

C

a. Copy and label the diagram. Include all the given information in your diagram.
b. Which angles do you know to be congruent? Explain.
c. Which sides do you know to be congruent? Explain.
d. Can you conclude that △ABC ≅ △AYX using the definition of congruent

triangles? If not, what additional information would you need?

224 Chapter 4  Congruent Triangles

Concept Byte Building Congruent MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Triangles
Use With Lesson 4-2 Prepares for GM-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce . . .
crointegrriuaefnocret r.i a. n. gcrleitsertioasfolrvteriparnogblleesmtos asonldve
Activity provbele rmelsaatinodnsphriopvsei nreglaetoiomnesthripc sfiignugreeso.metric
MfigPur3es.
MP 3

Can you use shortcuts to find congruent triangles? Find out by building and C
comparing triangles.
5013061020 70 80 90 100 110 6102050130
1 110 100 90 80 70

Step 1 Cut straws into three pieces of lengths 4 in., 5 in., and 6 in. Thread 1530014040 140401530016200
a string through the three pieces of straw. The straw pieces can be
in any order. 20
160
Step 2 Bring the two ends of the string together to make a triangle. Tie
the ends to hold your triangle in place. 10 170
170 10
Step 3 Compare your triangle with your classmates’ triangles. Try to
make your triangle fit exactly on top of the other triangles. 0 180
180 0
1. Is your triangle congruent to your classmates’ triangles?
A B
2. Make a Conjecture  What seems to be true about two triangles in
which three sides of one are congruent to three sides of another? hsm11gmse_04fa_t05011

3. As a class, choose three different lengths and repeat Steps 1–3. Are 5013061020 70 80 90 100 110 6102050130
all the triangles congruent? Does this support your conjecture from 110 100 90 80 70
Question 2?
1530014040 140401530016200
2
20
Step 1 Use a straightedge to draw and label any △ABC on tracing paper. 160

Step 2 Use a ruler. Carefully measure AB and AC. Use a protractor to measure the 10 170
angle between them, ∠A. 170 10

Step 3 Write the measurements on an index card and swap cards with a classmate. 0 180
Draw a triangle using only your classmate’s measurements. 180 0

Step 4 Compare your new triangle to your classmate’s original △ABC. Try to make A B
your classmate’s △ABC fit exactly on top of your new triangle.

4. Is your new triangle congruent to your classmate’s original △ABC?

5. Make a Conjecture  What seems to be true about two triangles when they have
two congruent sides and a congruent angle between them?

6. Make a Conjecture  At least how many triangle measurements must you know
in order to guarantee that all triangles built with those measurements will be
congruent?

hsm11gmse_04fa_t05013

Concept Byte  Building Congruent Triangles 225

4-2 Triangle Congruence MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
by SSS and SAS
GM-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce c.o. n. gcriuternicaefo. r. .trciraintegrlieasfor
troi asnoglvlesptroo bsolelvmesparnodblpermosveanredlaptriovneshrieplsatiinongsehoimpseitnric
gfiegoumrees.tric figures.

MP 1, MP 3, MP 4, MP 7

Objective To prove two triangles congruent using the SSS and SAS Postulates

Are the triangles below congruent? How do you know?

How can you tell 8y C
whether these 6 E
triangles are
congruent? In this B A
lesson, you will learn 4
the least amount
of information 2
required to tell if F Dx
two triangles are
congruent. O 2 4 6 8 10 12 14 16

MATHEMATICAL

PRACTICES
In the Solve It, you looked for relationships between corresponding sides and
angles. In Lesson 4-1, you learned that if two triangles have three pairs of congruent
corresponding angles and three pairs of congruent corresponding sides, then the
triangles are congruent.

If you know . . . FJ

∠F ≅ ∠J FG ≅ JK

∠G ≅ ∠K GH ≅ KL

∠H ≅ ∠L FH ≅ JL G HK L

. . . then you know △FGH ≅ △JKL.

However, this is more information about the corresponding parts than you need to

prove triangles congruent.

hsm11gmse_0402_t05014
Essential Understanding  You can prove that two triangles are congruent

without having to show that all corresponding parts are congruent. In this lesson,

you will prove triangles congruent by using (1) three pairs of corresponding sides

and (2) two pairs of corresponding sides and one pair of corresponding angles.

226 Chapter 4  Congruent Triangles

Postulate 4-1  Side-Side-Side (SSS) Postulate

Postulate If . . . Then . . .
If the three sides of one AB ≅ DE, BC ≅ EF , AC ≅ DF △ABC ≅ △DEF
triangle are congruent to
the three sides of another BE
triangle, then the two
triangles are congruent. AD

CF

As described in Chapter 1, a postulate is an accepted statement of fact. The Side-Side-
Side Postulate is perhaps the most logical fact about triangles. It agrees with the notion

that triangles are rigid figures; thhesimr s1ha1pgemdsoees_0no4t0c2h_atn0g5e0u1n7til pressure on their sides

forces them to break. This rigidity property is important to architects and engineers
when they build things such as bicycle frames and steel bridges.

Problem 1 Using SSS PN

Proof Given:  LM ≅ NP, LP ≅ NM
Prove:  △LMN ≅ △NPL
You have two pairs
of congruent sides. LM
What else do you
need?
You need a third pair of
congruent corresponding
sides. Notice that
the triangles share a
common side, LN.

LM ≅ NP LN ≅ LN LP ≅ NM
Given Reflexive Prop. of ≅ Given

△LMN ≅ △NPL C
F
SSS

Got It? 1. Given:  BC ≅ BF , CD ≅ FD B D

Prove:  △BCD ≅ △BFD

hsm11gmls_0402_t05019

Lesson 4-2  Triangle Congruence by SSS and SAS 227

You can also show relationships ∠A is included B BC is included
between a pair of corresponding between BA between ∠B
sides and an included angle. and AC. and ∠C.

The word included refers to the A C
angles and the sides of a triangle
as shown at the right.

Postulate 4-2  Side-Angle-Side (SAS) Postulate

Postulate If . . . hsm11gmse_0402_t0501T6hen . . .
If two sides and the
included angle of one AB ≅ DE, ∠A ≅ ∠D, △ABC ≅ △DEF
triangle are congruent to
two sides and the included AC ≅ DF
angle of another triangle,
then the two triangles are BE
congruent.
AD
CF

You likely have used the properties of the Side-Angle-Side Postulate before. For

example, SAS can help you determhisnme w1h1egtmhesrea_b0o4x0w2i_lltf0it5th0r2o0ugh a doorway.

Suppose you keep your arms at a fixed angle as you move from the box to the doorway.
The triangle you form with the box is congruent to the triangle you form with the
doorway. The two triangles are congruent because two sides and the included angle of
one triangle are congruent to the two sides and the included angle of the other triangle.

228 Chapter 4  Congruent Triangles

Do you need another Problem 2 Using SAS EF
pair of congruent
sides? What other information do you need to prove DG
Look at the diagram. △DEF @ △FGD by SAS? Explain. LE
The triangles share DF. The diagram shows that EF ≅ GD. Also, DF ≅ DF by
So, you already have two the Reflexive Property of Congruence. To prove that hsm11gmNse_0402_t05022 B
pairs of congruent sides. △DEF ≅ △FGD by SAS, you must have congruent
included angles. You need to know that ∠EFD ≅ ∠GDF.

Got It? 2. What other information do you need to prove
△LEB ≅ △BNL by SAS? hsm11gmse_0402_t05023
A BC D
Recall that, in Lesson 1-6, you learned to construct
segments using a compass open to a fixed angle. Now
you can show that it works. Similar to the situation with
the box and the doorway, the Side-Angle-Side Postulate
tells you that the triangles outlined at the right are
congruent. So, AB ≅ CD.

Problem 3 Identifying Congruent Triangles hsm11gmse_0402_t05024

Would you use SSS or SAS to prove the triangles congruent? If there is not enough

information to prove the triangles congruent by SSS or SAS, write not enough

What should you information. Explain your answer.

look for first, sides or A B
angles?

Start with sides. If you

have three pairs of

congruent sides, use SSS.
If you have two pairs of Use SAS because two pairs of There is not enough information; two pairs of

congruent sides, look corresponding sides and their included corresponding sides are congruent, but one

for a pair of congruent ahnsgmle1s 1argemcosneg_r0u4en0t2. _t05025 hosfmth1e1anggmlesseis_n0o4t0t2h_eti0nc5l0u2d6ed angle.
included angles.

C D

Use SSS because three pairs of Use SSS or SAS because all three pairs of

corresponding sides are congruent. corresponding sides and a pair of included

hsm11gmse_0402_t 05027 ahnsgmle1s 1(tghme vseer_ti0ca4l0a2n_glte0s5) 0ar2e8congruent.

Got It? 3. Would you use SSS or SAS to prove the triangles at the
right congruent? Explain.

Lesson 4-2  Triangle Congruence by SShSsamnd11SAgSm se_0402_t02520929

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. In △PEN, name the angle that is included between 5. C ompare and Contrast  How are the SSS Postulate

the given sides. and the SAS Postulate alike? How are they different?

a. PE and EN b. NP and PE 6. Error Analysis  Your friend thinks that the triangles
shown below are congruent by SAS. Is your friend
2. In △HAT, between which sides is the given angle correct? Explain.

included?

a. ∠H b. ∠T

Name the postulate you would use to prove the
triangles congruent.

3. 4. 7. Reasoning  A carpenter trims a triangular peak
of a house with three 7-ft pieces of molding. The

ctrairapnegnutleahrrsupmseea1sk12.g1Amrfteosthef me_0tow4ldo0itn2rg_iattno0g5tlre0ism3f2oarsmeecodnd

congruent? Explain.

hsm11gmse_0402_t0503h0sm11gmse_0402_t05031 MATHEMATICAL
Practice and Problem-Solving Exercises
PRACTICES

A Practice 8. Developing Proof  Copy and complete the M See Problem 1.
flow proof.
L
J
Given:  JK ≅ LM, JM ≅ LK
Prove:  △JKM ≅ △LMK K

JK ≅ LM JM ≅ LK KM ≅ KM
Given a. b.

hsm11gmse_0402_t02639.ai

c. ≅ d.
SSS

9. Given:  IE ≅ GH, EF ≅ HF, 10. Given:  WZ ≅ ZS ≅ SD ≅ DW
Proof F is the midpoint of GI P roo f Prove:  △WZD ≅ △SDZ
W
Phrosvme:1  1△gEmFIs≅e_△04H0F2G_t02640.ai
ZD
E G
F S

IH

hsm11gmse_0402_t02641.ai hsm11gmse_0402_t02642.ai

230 Chapter 4  Congruent Triangles

What other information, if any, do you need to prove the two triangles See Problem 2.
congruent by SAS? Explain. W

11. G N 12. T U
LT Q R S V
M

Would you use SSS or SAS to prove the triangles congruent? If there is not See Problem 3.

enough information to prove the triangles congruent by SSS or SAS, write not
hsm11gmse_0402_t02644.ai
enouhghsmin1fo1rgmmatsieo_n0. E4x0p2l_aitn02yo6u4r3a.aniswer.

13. P 14. A

R BD
Q

T C
S

B Apply 15. Think About a Plan  You and a friend are cutting triangles out of felt for an art

project. You want all the triangles to be congruent. Your friend tells you that each

trhiasnmgl1e1sghmouslde_h0av4e0t2w_ot05-2in6.4s5id.aeis and a 40° angle. If yhousmfo1ll1owgmthsiser_u0le4,0w2i_llta0ll2646.ai

your felt triangles be congruent? Explain.
• How can you use diagrams to help you?
• Which postulate, SSS or SAS, are you likely to apply to the given situation?

16. Given:  BC ≅ DA, ∠CBD ≅ ∠ADB 17. Given:  X is the midpoint of AG and NR.
Proo f Prove:  △BCD ≅ △DAB P roo f Prove:  △ANX ≅ △GRX

C D A

BA XR
N

G

Use the Distance Formula to determine whether △ABC and △DEF are

congruent. Jhusstmify1y1ogumr asen_sw0e4r0. 2_t02647.ai hsm1210g. mA(s2e,_90),4B0(22,_4t)0, 2C6(54,84.)a; i
18. A(1, 4), B(5, 5), C(2, 2); 19. A(3, 8), B(8, 12), C(10, 5);

D( -5, 1), E( -1, 0), F( -4, 3) D(3, -1), E(7, -7), F(12, -2) D(1, -3), E(1, 2), F( -2, 2)

21. Writing  List three real-life uses of congruent triangles. For each real-life use,
describe why you think congruence is necessary.

Lesson 4-2  Triangle Congruence by SSS and SAS 231

22. Sierpinski’s Triangle  Sierpinski’s triangle is a famous
geometric pattern. To draw Sierpinski’s triangle, start with
a single triangle and connect the midpoints of the sides
to draw a smaller triangle. If you repeat this pattern over
and over, you will form a figure like the one shown. This
particular figure started with an isosceles triangle. Are the
triangles outlined in red congruent? Explain.

23. Constructions  Use a straightedge to draw any triangle
JKL. Then construct △MNP ≅ △JKL using the
given postulate.

a. SSS
b. SAS

Can you prove the triangles congruent? If so, write the congruence statement
and name the postulate you would use. If not, write not enough information and
tell what other information you would need.

24. A G 25. Y K 26. J S

R T HP
N DT
E FV
W

27. Reasoning  Suppose GH ≅ JK , HI ≅ KL, and ∠I ≅ ∠L. Is △GHI congruhensmt to11gmse_0402_t02651.ai
△JKL? Explain.
28. Ghivsemn:1  1GgKmbsisee_c0ts4∠02JG_Mt0,2G6J4≅9.GaiM hsm11gms e2_9.0 G40iv2e_nt: 0A2E65an0d.aBi D bisect each other.
Proo f Prove:  △GJK ≅ △GMK P roo f Prove: △ACB ≅ △ECD

J A B

GK C

M
DE

30. Given:  FG } KL, FG ≅ KL 31. Given:  AB # CM, AB # DB, CM ≅ DB,
Proof
Proo f Prove:  △hsFmGK11≅gm△KseLF_0402_t02652.ai M is the midpoint of AB
Prove:  △hAsmM1C1≅gm△sMeB_D0402_t02653.ai
F G

D B

LK C
M

A

hsm11gmse_0402_t02654.ai hsm11gmse_0402_t02655.ai

232 Chapter 4  Congruent Triangles

C Challenge 32. Given:  HK ≅ LG, HF ≅ LJ , FG ≅ JK 33. Given:  ∠N ≅ ∠L, MN ≅ OL, NO ≅ LM
P roo f Prove:  MN } OL
Proo f Prove:  △FGH ≅ △JKL
F M

HK GL NL
J O

34. Reasoning  Four sides of polygon ABCD are congruent, respectively, to the four
sides of polygon EFGH. Are ABCD and EFGH congruent? Is a quadrilateral a rigid

figuhres?mIf1n1ogt,mwhsaet_c0o4u0ld2y_otu02ad6d56to.ami ake it a rigid figure? Exphlasimn.11gmse_0402_t02657.ai

Standardized Test Prep

SAT/ACT 35. What additional information do you need to prove that Y
△VWY ≅ △VWZ by SAS?

YW ≅ ZW ∠Y ≅ ∠Z V W

∠WVY ≅ ∠WVZ VZ ≅ VY

36. The measures of two angles of a triangle are 43 and 38. What is the Z
measure of the third angle?

Short 9 81 99 100

Response 37. Which method would you use to find the inverse of a conditional statemehnstm? 11gmse_0402_t02658.ai

Switch the hypothesis and conclusion. Negate the conclusion only.

Negate the hypothesis only. Negate both the hypothesis and conclusion.

38. A segment has a midpoint at (1, 1) and an endpoint at ( -3, 4). What are the
coordinates of the other endpoint of the segment? Show your work.

Mixed Review

ABCD @ EFGH. Name the angle or side that corresponds to each part. See Lesson 4-1.
42. ∠G
39. ∠A 40. EF 41. BC

Write the converse of each statement. Determine whether the statement and its See Lesson 2-2.
converse are true or false.

43. If x = 3, then 2x = 6. 44. If x = 3, then x2 = 9.

G et Ready!  To prepare for Lesson 4-3, do Exercises 45 and 46. See Lesson 4-2.
45. In △JHK, name the side that is included between ∠J and ∠H.
46. In △NLM, name the angle that is included between NM and LN.

Lesson 4-2  Triangle Congruence by SSS and SAS 233

4-3 Triangle Congruence MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
by ASA and AAS
MG-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce c.o. n. gcriuternicaefo. r. .trciraintegrlieasftoor
tsroialvnegplersobtolesmoslvaenpdropbrolevme srealnatdiopnrsohviepsreilnatgioeonmsheiptrsicin
gfiegoumrees.tric figures.

MP 1, MP 3, MP 7

Objective To prove two triangles congruent using the ASA Postulate and the AAS Theorem

Oh no! The school’s photocopier is not working correctly. The copies all
have some ink missing. Below are two photocopies of the same geometry
worksheet. Which triangles are congruent? How do you know?

Use what you
already know about
proving triangles
congruent. What
is your plan for
finding an answer?

MATHEMATICAL

PRACTICES

You already know that triangles are congruent if two pairs of sides and the included
angles are congruent (SAS). You can also prove triangles congruent using other
groupings of angles and sides.

Essential Understanding  You can prove that two triangles are congruent
without having to show that all corresponding parts are congruent. In this lesson, you
will prove triangles congruent by using one pair of corresponding sides and two pairs
of corresponding angles.

Postulate 4-3  Angle-Side-Angle (ASA) Postulate

Postulate If . . . E Then . . .
If two angles and the ∠A ≅ ∠D, AC ≅ DF , △ABC ≅ △DEF
included side of one triangle ∠C ≅ ∠F
are congruent to two angles F
and the included side of B
another triangle, then the
two triangles are congruent. AC
D

234 Chapter 4  Congruent Triangles hsm11gmse_0403_t05033.ai

Problem 1 Using ASA ON
Which two triangles are congruent by ASA? Explain.

From the diagram you know U E W
• ∠U ≅ ∠E ≅ ∠T S VT A
• ∠V ≅ ∠O ≅ ∠W
• UV ≅ EO ≅ AW

To use ASA, you need two pairs You already have pairs of congruent angles. So,
of congruent angles and a pair of identify the included side for each triangle and see
included congruent sides.
whether it has a congruehncsemma1r1kigngm. se_0403_t05036.ai

In △SUV, UV is included between ∠U and ∠V and has a congruence marking. In
△NEO, EO is included between ∠E and ∠O and has a congruence marking. In △ATW,
TW is included between ∠T and ∠W but does not have a congruence marking.

Since ∠U ≅ ∠E, UV ≅ EO, and ∠V ≅ ∠O, △SUV ≅ △NEO.

Got It? 1. Which two triangles are congruent H FC
GN IT
by ASA? Explain. O A

Proof Problem 2 Writing a Proof Using ASA hsm11gmse_0403_t05034.ai

Can you use a plan Recreation  Members of a teen organization are building
similar to the plan in
Problem 1? a miniature golf course at your town’s youth center. The
Yes. Use the diagram to
identify the included side design plan calls for the first hole to have two congruent
for the marked angles in
each triangle. triangular bumpers. Prove that the bumpers on the first A D
hole, shown at the right, meet the conditions of the plan. F

Given:  AB ≅ DE, ∠A ≅ ∠D, ∠B and ∠E are right angles BE

Prove:  △ABC ≅ △DEF C

Proof:   ∠B ≅ ∠E because all right angles are congruent,
and you are given that ∠A ≅ ∠D. AB and DE are
included sides between the two pairs of congruent

angles. You are given that AB ≅ DE. Thus,
△ABC ≅ △DEF by ASA.

Lesson 4-3  Triangle Congruence by ASA and AAS 235

Got It? 2. Given:  ∠CAB ≅ ∠DAE, BA ≅ EA, CD
∠B and ∠E are right angles
A
Prove:  △ABC ≅ △AED BE

You can also prove triangles congruent by using two angles and a nonincluded side, as

stated in the theorem below. hsm11gmse_0403_t05037.ai

Theorem 4-2  Angle-Angle-Side (AAS) Theorem

Theorem If . . . Then . . .
If two angles and a ∠A ≅ ∠D, ∠B ≅ ∠E, △ABC ≅ △DEF
nonincluded side of one AC ≅ DF
triangle are congruent to two F
angles and the corresponding B
nonincluded side of another E
triangle, then the triangles are C
congruent.
AD

Proof Proof of Theorem 4-2:  Angle-Angle-Side Theorem B
Given:  ∠A ≅ ∠D, ∠B ≅ ∠E, AC ≅ DhsFm11gmse_0403_t05038.ai

Prove:  △ABC ≅ △DEF CF

∠A ≅ ∠D ∠C ≅ ∠F AD
Given Third Angles Theorem E

∠B ≅ ∠E ᭝ABC ≅ ᭝DEF hsm11gmse_0403_t05039.ai
Given ASA

AC ≅ DF
Given

Yahnosudmhflao1vw1egpsmreoeosnef.a_En0ad4cuh0s3me_dettt0hh5ored0e4ims0e.eaqtihuoaldlys of proof in this book—two-column, paragraph,
as valid as the others. Unless told otherwise,

you can choose any of the three methods to write a proof. Just be sure your proof always

presents logical reasoning with justification.

236 Chapter 4  Congruent Triangles

Proof Problem 3 Writing a Proof Using AAS M R
W K
How does information Given:  ∠M ≅ ∠K, WM } RK
about parallel sides Prove:  △WMR ≅ △RKW
help?
You will need another Statements Reasons
pair of congruent angles
to use AAS. Think back 1) ∠M ≅ ∠K 1) Given
to what you learned
in Chapter 3. WR is a 2) WM } RK 2) Given hsm11gmse_0403_t05041.ai
transversal here. 3) ∠MWR ≅ ∠KRW
4) WR ≅ WR 3) If lines are }, then alternate interior ⦞ are ≅.
5) △WMR ≅ △RKW 4) Reflexive Property of Congruence

5) AAS

Got It? 3. a. Given:  ∠S ≅ ∠Q, RP bisects ∠SRQ S P Q
R
Prove:  △SRP ≅ △QRP

b. Reasoning  In Problem 3, how could you prove
that △WMR ≅ △RKW by ASA? Explain.

Problem 4 Determining Whether Triangles Are Conhgsmru1e1ngtmse_0403_t05042.ai
B
Multiple Choice  Use the diagram at the right. Which of the

following statements best represents the answer and justification

to the question, “Is △BIF @ △UTO?”

Yes, the triangles are congruent by ASA.

Can you eliminate No, FB and OT are not corresponding sides. F I
any  of the choices? U
Yes. If △BIF @ △UTO Yes, the triangles are congruent by AAS.
then ∠B and ∠U would
be corresponding angles. No, ∠B and ∠U are not corresponding angles.

You can eliminate The diagram shows that two pairs of angles and one pair of sides
are congruent. The third pair of angles is congruent by the Third
choice D. Angles Theorem. To prove these triangles congruent, you need to
satisfy ASA or AAS.
OT
ASA and AAS both fail because FB and TO are not included
between the same pair of congruent corresponding angles, so they hsm11gmse_0403_t05044.ai
are not corresponding sides. The triangles are not necessarily I
congruent. The correct answer is B.

Got It? 4. Are △PAR and △SIR congruent? Explain. P

R
AS

Lesson 4-3  Triangle Congruence by ASA and AAS 237

hsm11gmse_0403_t05045.ai

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. In △RST, which side is included between
5. Compare and Contrast  How are the ASA Postulate
∠R and ∠S?
and the SAS Postulate alike? How are they different?
2. In △NOM, NO is included between which angles?
6. Error Analysis  Your friend asks you for help on a
Which postulate or theorem could you use to prove geometry exercise. Below is your friend’s paper. What
△ABC @ △DEF ? error did your friend make? Explain.

3. A D M S ∆LMN ≅ ∆QRS
LN Q by ASA.
CF R

BE

4. A B E D 7. Reasoning  Suppose ∠E ≅ ∠I and FE ≅ GI .

hsm11gmse_0C403F _t05046.ai What else must you know in order to prove
h△smFD11Eg≅ms△eG_H04I0b3y_At0S5A0?4B8y.aAiAS?

hsm11gmse_0403_t05047.ai MATHEMATICAL

Practice and Problem-Solving Exercises PRACTICES

A Practice Name two triangles that are congruent by ASA. See Problem 1.

8. P Q S X W 9. E F
A B
H
U D
R
TV CG I

10. Developing Proof  Complete the paragraph proof by filling in the blanks. See Problem 2.

Given:  ∠LKM ≅ ∠JKM, K hsm11gmse_0403_t02691.ai
J
hsm11g∠mLsMeK_0≅40∠3_JMt0K2690.ai

Prove:  △LKM ≅ △JKM LM

Proof:  ∠LKM ≅ ∠JKM and

∠LMK ≅ ∠JMK are given. KM ≅ KM by the

a.  ?  Property of Congruence. So, △LKM ≅ △JKM by b. ? .

11. Given:  ∠BAC ≅ ∠DAC, hsmB 11g 1m2.s Ge_iv0e4n0: 3Q_tR02≅69T2S.,ai Q R
Proof C Proof S T
AC # BD A QR } TS

Prove:  △ABC ≅ △ADC D Prove: △QRT ≅ △TSQ

hsm11gmse_0403_t02693.ai hsm11gmse_0403_t02694.ai

238 Chapter 4  Congruent Triangles

13. Developing Proof  Complete the two-column proof by filling in the blanks. See Problem 3.

Given:  ∠N ≅ ∠S, N ᐍ R
line / bisects TR at Q Q S
T
Prove:  △NQT ≅ △SQR Reasons

Statements

1) ∠N ≅ ∠S 1) Given
2) ∠NQT ≅ ∠SQR 23)) abh..sm??11gmse_0403_t02695.ai

3) Line / bisects TR at Q. 4) Definition of bisect
4) c. ? 5) d. ?
5) △NQT ≅ △SQR

14. Given:  ∠V ≅ ∠Y, 15. Given:  PQ # QS, RS # SQ,
Proo f WZ bisects ∠VWY Proof T is the midpoint of PR
Prove:  △VWZ ≅ △YWZ
Prove:  △PQT ≅ △RST
W R

VZY Q

TS
P

Determine whether the triangles must be congruent. If so, name the postulate See Problem 4.

or theorem that justifies your answer. If not, explain.

16h.s m 11gmse_0M403_t02696.ai 17. T hsm11gms1e8_. 0403V_t02697.aiW

U

P ON R
SZ
Y

B Apply 19. Given:  ∠N ≅ ∠P, MO ≅ QO 20. Given:  ∠FJG ≅ ∠HGJ, FG } JH
Proo f Phrsomv1e1: g△mMseO_N04≅03△_Qt0O2P69 8.ai hsm11gmsPero_o0f 4P0ro3v_et:0  2△6F9G9J.a≅i
△HJG
M F
N hsGm11gmse_0403_t02700

O JH
P

Q

hsm11gmse_0403_t02701.ai hsm11gmse_0403_t02702.ai

Lesson 4-3  Triangle Congruence by ASA and AAS 239

21. Think About a Plan  While helping your family clean out
the attic, you find the piece of paper shown at the right.
The paper contains clues to locate a time capsule buried
in your backyard. The maple tree is due east of the oak
tree in your backyard. Will the clues always lead you to
the correct spot? Explain.

• How can you use a diagram to help you?
• What type of geometric figure do the paths and the

marked line form?
• How does the position of the marked line relate to the

positions of the angles?

22. Constructions  Use a straightedge to draw a triangle. Label
it △JKL. Construct △MNP ≅ △JKL so that the triangles are

congruent by ASA.

23. Reasoning  Can you prove that the triangles at the right are congruent? Justify your

answer. HSM11GMSE_0403_a02291

24. Writing  Anita says that you can rewrite any proof that uses the AAS Theor3ermd paassas 12-22-08

proof that uses the ASA Postulate. Do you agree with Anita? Explain. Durke

25. Given:  AE } BD, AE ≅ BD, 26. Given:  ∠1 ≅ ∠2, and
Proo f ∠E ≅ ∠D Proof
Prove:  △AEB ≅ △BDC DH bisects ∠BDF. hsm11gmse_0403_t02703

ED Prove:  △BDH ≅ △FDH
D

A BC 12
H
BF

27. Draw a Diagram  Draw two noncongruent triangles that have two pairs of

congruent angles and one pair of congruent sides.

Pr2o8o .f PGhriovsevmne::1   1△AgBAmB}CseD≅_C0,△4AC0D3D_}AtB0C2704.ai A hsm11gmsBe_0403_t02706

DC

C Challenge 29. Given AD } BC and AB } DC, name as many pairs of congruent B C

triangles as you can. E

30. Constructions  In △RST at the right, hsm1R1gmse_0403_t02708.ai D
RS = 5, RT = 9, and m∠T = 30. Show
that there is no SSA congruence rule by A

constructing △UVW with UV = RS, 9
UW = RT , and m∠W = m∠T , but with 5
△UVW R △RST .
S 30Њ hsmT11gmse_0403_t02709.ai

240 Chapter 4  Congruent Triangles hsm11gmse_0403_t02710.ai

31. Probability  Below are six statements about the triangles at the right. A
XB
∠A ≅ ∠X ∠B ≅ ∠Y ∠C ≅ ∠Z
Y
AB ≅ XY AC ≅ XZ BC ≅ YZ C
Z
There are 20 ways to choose a group of three statements from these six.
What is the probability that three statements chosen at random from the
six will guarantee that the triangles are congruent?

Standardized Test Prep

SAT/ACT 32. Suppose RT ≅ ND and ∠R ≅ ∠N. What additional information do you need thosm11gmse_0403_t02720

prove that △RTJ ≅ △NDF by ASA?

∠T ≅ ∠D ∠J ≅ ∠F ∠J ≅ ∠D ∠T ≅ ∠F

33. You plan to make a 2 ft-by-3 ft rectangular poster of class trip photos. Each photo is
a 4 in.-by-6 in. rectangle. If the photos do not overlap, what is the greatest number
of photos you can fit on your poster?

4 24 32 36

34. Which of the following figures is a concave polygon?


Short 35. Write the converse of the true conditional statement below. Then determine
Response whether the converse is true or false.

If yhousmar1e1legsms tshea_n01480y3ea_rts0ho2sld7m,2t11h1egnmyosuea_r0e4to0o3y_otu0n2hg7stm2o2v1o1tgeminstehe_0U4n0it3ed_tS0tha2ts7em2s.311gmse_0403_t05009

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 3

Look back at the information given on page 217 about how Jamal located

the points in the diagram. In the Apply What You’ve Learned in Lesson 4-1,

you copied the diagram, labeled it with the given information, and identified

congruent sides and angles.

a. Look at the diagram you labeled. Which congruence postulate or theorem
can you use to prove the two triangles are congruent?

b. Write a proof that the two triangles are congruent using only the information that
you already have.

Lesson 4-3  Triangle Congruence by ASA and AAS 241

Concept Byte Exploring AAA MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and SSA
Use With Lesson 4-3 Extends GM-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce c.o. n. gcriuternicae . . .
cforirt etrianfgolre tsritaonsgolelvsetoprsoobllveempsroabnldemprsoavnedreplarotivoenships in
TECHNOLOGY greeloamtioentrsihcipfisguinregse. ometric figures.

MP 5

So far, you know four ways to conclude that two triangles are congruent—SSS, SAS, MATHEMATICAL
ASA, and AAS. It is good mathematics to wonder about the other two possibilities.
PRACTICES

1
Cfoornmst△ruAcBt CU. sCeognesotrmucettraylsinoeftwpaarraellteolctoonBsCtruthcat tAiBn>tearnsdecAtsC>A. CB>oannsdtruAcCt>
BC to
at

points D and E to form △ADE. B
C
Investigate  Are the three angles of △ABC congruent to the three angles of A D
△ADE? Manipulate the figure to change the positions of DE and BC. Do the E
corresponding angles of the triangles remain congruent? Are the two triangles

congruent? Can the two triangles be congruent?

2

Construct Construct AB>. Draw a circle with center C that intersects AB> at two

points. Construct AC. Construct point E on the circle and construct CE. hsm11gmse_04fa_t05104.ai
pEooinntthEeacroirucnledttohtehceiroctlheeurnptoilinEtiosnonABA>Bt>oafnodrmforms
Investigate  Move D
△ACE. Then move
C
another △ACE. E

Compare AC, CE, and m∠A in the two triangles. Are two sides and a AB
nonincluded angle of one triangle congruent to two sides and a nonincluded

angle of the other triangle? Are the triangles congruent? If you change the
measure of ∠A and the size of the circle, do you get the same results?

Exercises hsm11gmse_04fa_t05105.ai

1. Make a Conjecture  Based on your first investigation above, can you prove
triangles congruent using AAA? Explain.

For Exercises 2–4, use what you learned in your second investigation above.

2. Make a Conjecture  Can you prove triangles congruent using SSA? Explain.
3. Manipulate the figure so that ∠A is obtuse. Can the circle intersect AB> twice to

form two triangles? Would SSA work if the congruent angles were obtuse? Explain.

4. Suppose you are given CE, AC, and ∠A. What must be true about CE, AC, and
m∠A so that you can construct exactly one △ACE? (Hint: Consider cases.)

242 Concept Byte  Exploring AAA and SSA

4 Mid-Chapter Quiz M athX

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

FO

Do you know HOW? Determine what other information you need to
prove the two triangles congruent. Then write the
1. △RST ≅ △JKL. List the three pairs of congruent congruence statement and name the postulate or
corresponding sides and the three pairs of congruent theorem you would use.
corresponding angles.

LMNO @ PQRS. Name the angle or side that 15. A B 16. A D
corresponds to the given part.
O
2. RS 3. ∠L DC B C F
4. ∠Q 5. MN E

17. Constructions  Construct △LMN Q

State the postulate or theorem you can use to prove congruent to△PQR using SSS.
the triangles congruent. If you do not have enough hsm11gmse_04mq_t0259h5s.mai 1P1gmse_04Rmq_t02597.ai
information to prove the triangles congruent, write not
enough information. Do you UNDERSTAND?

6. 7. 18. Given:  ∠A ≅ ∠D, O is the midpoint of AD

Prove:  △AOB ≅ △DOC hsm11gmse_04mq_t026

A C

8. 9. BOD

hsm11gmse_04mq_t02592.ai 19. Given:  AB ≅ BC ≅ CD ≅ DA,

10. 11. hsm11gmse_04mq_t02593.ai ∠A, ∠B, ∠C, and ∠D are right angles

Prove:  △ABC ≅hs△mC1D1Agmse_04mq_t02600

AB

hsm11gmse_04mq_t0259h4s.mai11gmse_04mq_t02596.ai

Use the diagram below. Tell why each statement is true. DC

12. ∠∠hsHHmN≅1L1∠≅gKm∠KseN_J04mq_t0259h8sLm11gNmse_04mKq_t02599 20. Reasoning  Three segments form a triangle. How
13.
14. △HNL ≅ △KNJ many unique triangles can you construct using the

same three segments? Using  the same three angles?
hsm11gmse_04mq_t02603
Explain.

H 21. Open-Ended  Write a congruency postulate for
J quadrilaterals. Does your postulate always hold

true? Explain.

hsm11gmse_04mq_t02601 243

Chapter 4  Mid-Chapter Quiz

4-4 Using Corresponding Parts CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
of Congruent Triangles
MG-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce c.o. n. gcriuternicae . . .
fcorirtetrianfgolretsritaonsgolelvsetoprsoobllveempsroabnldemprsoavned
rperloavteiorneslhaitpiosnisnhgipesominegteriocmfiegturricesf.igAulrseos. Also
GM-ACFOS.D.9.12.G-SRT.4.12

MP 1, MP 3

Objective To use triangle congruence and corresponding parts of congruent triangles
to prove that parts of two triangles are congruent

Is △ABC congruent to △GHI? How do you know?

How does △DEF
help you solve this
problem?

MATHEMATICAL

PRACTICES

With SSS, SAS, ASA, and AAS, you know how to use three congruent parts of two
triangles to show that the triangles are congruent. Once you know that two triangles are
congruent, you can make conclusions about their other corresponding parts because,
by definition, corresponding parts of congruent triangles are congruent.

Essential Understanding  If you know two triangles are congruent, then you
know that every pair of their corresponding parts is also congruent.

Proof Problem 1 Proving Parts of Triangles Congruent K C
Given: ∠KBC ≅ ∠ACB, ∠K ≅ ∠A B A
In the diagram, which Prove: KB ≅ AC
congruent pair is not
marked? ЄKBC Х ЄACB BC Х BC
The third angles of both Given Reflexive Property of Х
triangles are congruent.
But there is no AAA ЄK Х ЄA ᭝KBC Х ᭝ACB KB Х AC
congruence rule. So, find Given AAS Theorem
a congruent pair of sides. Corresp. parts of Х hsm11gmse_0404_t02447

are Х.

Got It? 1. Given:  BA ≅ DA, CA ≅ EA CE
Prove:  ∠C ≅ ∠E A

hsm11gmse_0404_t02448 BD

244 Chapter 4  Congruent Triangles

Proof Problem 2 Proving Triangle Parts Congruent to Measure Distance STEM

Which congruency Measurement  Thales, a Greek philosopher, is said to have developed a method
rule can you use? to measure the distance to a ship at sea. He made a compass by nailing two sticks
You have information together. Standing on top of a tower, he would hold one stick vertical and tilt the
about two pairs of other until he could see the ship S along the line of the tilted stick. With this compass
angles. Guess-and- setting, he would find a landmark L on the shore along the line of the tilted stick.
check AAS and ASA. How far would the ship be from the base of the tower?

Given:  ∠TRS and ∠TRL are right angles, ∠RTS ≅ ∠RTL

Prove:  RS ≅ RL

T

SR L

Statements Reasons

1) ∠RTS ≅ ∠RTL 1) Given
2) TR ≅ TR 2) Reflexive Property of Congruence
3) ∠TRS and ∠TRL are right angles. 3) Given
4) ∠TRS ≅ ∠TRL 4) All right angles are congruent.
5) △TRS ≅ △TRL 5) ASA Postulate
6) RS ≅ RL 6) Corresponding parts of ≅ △s are ≅.

The distance between the ship and the base of the tower would be the same as the
distance between the base of the tower and the landmark.

Got It? 2. a. Given:  AB ≅ AC, M is the midpoint of BC B A
Prove:  ∠AMB ≅ ∠AMC M
b. Reasoning  If the landmark were not at sea level, C

would the method in Problem 2 work? Explain.

Lesson 4-4  Using Corresponding Parts of Congruent Triangles 245

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

Name the postulate or theorem that you can use to PRACTICES
show the triangles are congruent. Then explain why the
statement is true. 3. Reasoning  How does the fact that corresponding

parts of congruent triangles are congruent relate to

the definition of congruent triangles?

1. EA ≅ MA T 4. Error Analysis  Find and K M
H
correct the error(s) in
N
the proof. L

E AM Given:  KH ≅ NH, ∠L ≅ ∠M
U
2. ∠U ≅ ∠E Prove:  H is the midpoint of LM.
JN
Proof:  KH ≅ NH because it is given. ∠L ≅ ∠M
hsm11gmse_0404_t02450
because it is given. ∠KHL ≅h∠smNH11Mgbmescaeu_s0e4v0e4rt_icta0l2453
E
angles are congruent. So, △KHL ≅ △MHN by ASA
Postulate. Since corresponding parts of congruent
triangles are congruent, LH ≅ MH. By the definition
of midpoint, H is the midpoint of LM.

hsm11gmse_0404_t02452 MATHEMATICAL M
Practice and Problem-Solving Exercises
PRACTICES
L

A Practice 5. Developing Proof  Tell why the two triangles are See Problem 1.
congruent. Give the congruence statement. Then See Problem 2.
list all the other corresponding parts of the triangles K O
that are congruent. JN

6. Given:  ∠ABD ≅ ∠CBD, 7. Given:  OM ≅ ER, ME ≅ RO
Proof P roo f Prove:  ∠M ≅ ∠R
∠BDA ≅ ∠BDC
hsmO 11gmse_0404_t02457
Prove:  AB ≅ CB

B

A CM R

E
D

8. Developing Proof  A balalaika is a stringed instrument. Prove

that the bases of the balalaikas are congruent.

Ghivsemn1: 1RgAm≅seN_Y0,4∠0K4R_At0≅24∠6J1NY , ∠KAR ≅ ∠hJsYmN11gmse_0404_t02463

Prove:  KA ≅ JY R N
Proof:  It is given that two angles and the included side of one Y

triangle are congruent to two angles and the included side of the

other. So, a. ? ≅ △JNY by b. ? . KA ≅ JY because c. ? .

K A J

246 Chapter 4  Congruent Triangles

B Apply 9. Given:  ∠SPT ≅ ∠OPT , 10. Given:  YT ≅ YP, ∠C ≅ ∠R,
Proof
SP ≅ OP Proof ∠T ≅ ∠P

Prove:  ∠S ≅ ∠O Prove:  CT ≅ RP

S T O C R

Y

P TP

Reasoning Copy and mark the figure to show the given information. K
Explain how you would prove jP @ jQ.

11. Ghisvmen1: 1PgKm≅seQ_K04, K0L4_btis0e2c4ts6∠4PKQ hsm11gmse_0404_t02465

12. Given:  KL is the perpendicular bisector of PQ.

13. Given:  KL # PQ, KL bisects ∠PKQ P LQ

14. Think About a Plan  The construction of a line perpendicular to line / <CP> C
through point P on line / is shown. Explain why you can conclude that

is perpendicular to /. ᐉ
• How can you use congruent triangles to justify the construction?
hsm11gmse_04A04_t024P66
• Which lengths or distances are equal by construction? B

15. Given:  BA ≅ BC, BD bisects ∠ABC 16. Given:  / # AB, / bisects AB at C,
Proo f Prove:  BD # AC, BD bisects AC Proof
B P is on /

Prove:  PA = PB hsm11gmse_0404_t0246


P

A DC

A CB
ᐉ

17h. Csmon1st1rgumctisoen_s0  T4h0e4c_otn0s2tr4u7c1tion of ∠B congruent to given CE

∠A is shown. AD ≅ BF because they are congruent radii. hsm11gmse_0404_t02473
DC ≅ FE because both arcs have the same compass settings.

Explain why you can conclude that ∠A ≅ ∠B.

A DB F

18. Given:  BE # AC, DF # AC, 19. Given:  JK } QP, JK ≅ PQ
Proof P roo f Prove:  KQ bisects JP.
BE ≅ DF , AF ≅ CE

Prove:  AB ≅ CD K hsm11gmse_P0404_t02474

B C

EF JM Q

AD

hsm11gmse_0404_t02476 Lesson 4-4  UsihngsmCo1r1regspmonsdei_n0g4P0ar4ts_ot0f C2o4n7g9ruent Triangles 247

20. Designs  Rangoli is a colorful design pattern drawn outside houses
in India, especially during festivals. Vina plans to use the pattern at

the right as the base of her design. In this pattern, RU , SV , and QT
bisect each other at O. RS = 6, RU = 12, RU ≅ SV , ST } RU , and
RS } QT . What is the perimeter of the hexagon?

C Challenge In the diagram at the right, BA @ KA and BE @ KE. K

21. Prove:  S is the midpoint of BK . A S E
B
Proof

22. Prove:  BK # AE

Proof

PERFORMANCE TASK

Apply What You’ve Learned hsm11gmse_0404_t0248PM1RATAHCEMTAITCICEASL

MP 7

Look back at the information on page 217 and at your work from the Apply

What You’ve Learned sections in Lessons 4-1 and 4-3. Choose from the following

words and names of figures to complete the sentences below.

congruent corresponding AY AC

∠Y ∠YAX ∠X ∠C

△AYX △YAX BC YX

In the Apply What You’ve Learned in Lesson 4-3, you proved that △BAC ≅ a. ? .

Now, you can conclude that ∠C ≅ b. ? because c. ? parts of d. ? triangles
are congruent.

Similarly, e. ? and AX are congruent corresponding sides. Another pair of
congruent corresponding sides are f. ? and YX .

248 Chapter 4  Congruent Triangles

Concept Byte Paper-Folding CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Conjectures
Use With Lesson 4-5 MG-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl gaekoemfoertmricalcognesotmruecttriiocns with a
cvoanrisettryuoctfiotonoslswaitnhd amveatrhieotdyso(f. t.o.oplaspaenrdfomldeitnhgod. s. .).
Activity (M. .P. p3aper folding . . .).
MP 3

Isosceles triangles have two congruent sides. Folding one of the sides onto the other
will suggest another important property of isosceles triangles.

1

Step 1 Construct an isosceles △ABC on tracing paper, with AC ≅ BC.


CC

A BA BA B

Step 2 Fold the paper so the two congruent sides fit exactly one on top of the other. B

Crehassme t1he1gpampseer._L0a4befal _thte0i2n5te8r7se.acition of the fold line and AB as point D. A

1. What do you notice about ∠A and ∠B? Compare your results with others’. Make a
conjecture about the angles opposite the congruent sides in an isosceles triangle.

2. a. Study the fold line CD and the base AB. What type of angles are ∠CDA and ∠CDB?
How do AD and BD seem to be related?
b. Use your answers to part (a) to complete the conjecture:
The fold line CD is the ? of the base AB of isosceles △ABC.

2 hsm11gmse_04fa_t02588.ai

In Activity 1, you made a conjecture about angles opposite the congruent sides of a FG
triangle. You can also fold paper to study whether the converse is true. H

Step 1 On tracing paper, draw acute angle F and one side FG. Construct ∠G as shown, 12
so that ∠G ≅ ∠F .
hsm11gmse_04fa_t02589.ai
Step 2 Fold the paper so ∠F and ∠G fit exactly one on top of the other.
3. Why do sides 1 and 2 meet at point H on the fold line? Make a conjecture about FG

sides FH and GH opposite congruent angles in a triangle.

4. Write your conjectures from Questions 1 and 3 as a biconditional.

hsm11gmse_04fa_t02591.ai

Concept Byte  Paper-Folding Conjectures 249

4-5 Isosceles and MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Equilateral Triangles
GM-ACFOS.C.9.120. GP-rCovOe.3th.1e0o rePmrosvaebtohuetotrreimansgalebsou. .t . base
atrniagnlegsleosf.is.o. sbcaeslesatnrgialensgloefsisaorescceolensgrturieanntg.le.s. aArleso
cGo-nCgOru.Den.1t3.,. G. A-SlsRoT.MB.A5FS.912.G-CO.4.13,
MMAPF1S,.M91P2.3G, -MSRPT4.2.5
MP 1, MP 3, MP 4

Objective To use and apply properties of isosceles and equilateral triangles

Solving puzzles The triangles of the same color are congruent. Arrange the triangles
is fun! Work with to form one large triangle. You must use all the pieces. Make a sketch
the pieces until of this triangle. Classify this triangle by its sides. What are the angle
they make a whole measures of this triangle? Explain.
triangle. Look for
patterns in your 3
solution.
40؇ 50؇
MATHEMATICAL
6
PRACTICES

In the Solve It, you classified a triangle based on the lengths of its sides. You can also
identify certain triangles based on information about their angles. In this lesson, you
will learn how to use and apply properties of isosceles and equilateral triangles.

Essential Understanding  The angles and sides of isosceles and equilateral

triangles have special relationships. Vertex angle

Lesson Isosceles triangles are common in the real world. You can frequently

Vocabulary see them in structures such as bridges and buildings, as well as in art Legs
• legs of an and design. The congruent sides of an isosceles triangle are its legs. Base
The third side is the base. The two congruent legs form the vertex Base angles
isosceles triangle angle. The other two angles are the base angles.
• base of an
Theorem 4-3  Isosceles Triangle Theorem
isosceles triangle
• vertex angle Theorem If . . . Thehns.m. .11gmse_0405_t02725
If two sides of a triangle are AC ≅ BC
of an isosceles congruent, then the angles ∠A ≅ ∠B
triangle opposite those sides are
• base angles of an congruent.
isosceles triangle
• corollary

C C
AB
AB

250 Chapter 4  Congruent Triangles hsm11gmse_0405_t02726 hsm11gmse_0405_t02727

The proof of the Isosceles Triangle Theorem requires an auxiliary line.

Proof Proof of Theorem 4-3: Isosceles Triangle Theorem X

Begin with isosceles △XYZ with XY ≅ XZ. Draw XB, the bisector of 12
the vertex angle ∠YXZ.

Given: XY ≅ XZ, XB bisects ∠YXZ YBZ
Prove: ∠Y ≅ ∠Z

Proof: XY ≅ XZ is given. By the definition of angle bisector, ∠1 ≅ ∠2. By the Reflexive

Property of Congruence, XB ≅ XB. So by the SAS Postulate, a△rXe YcoBnhg≅srmu△e1nX1tZ.gBm. se_0405_t02728
∠Y ≅ ∠Z since corresponding parts of congruent triangles

Theorem 4-4  Converse of the Isosceles Triangle Theorem

Theorem If . . . Then . . .
If two angles of a triangle ∠A ≅ ∠B
are congruent, then the C AC ≅ BC C
sides opposite those angles
are congruent. AB AB

You will prove Theorem 4-4 in Exercise 23.

hsm11gmse_0405_t02729hsm11gmse_0405_t0

Problem 1 Using the Isosceles Triangle Theorems

A Is AB congruent to CB? Explain. B
E
W hat are you looking Yes. Since ∠C ≅ ∠A, AB ≅ CB by the Converse of the

for in the diagram? Isosceles Triangle Theorem. D
A
To use the Isosceles B Is jA congruent to jDEA? Explain.
T riangle Theorems, you

an neegdleas opraair poaf icroonfgruent Yes. Since AD ≅ ED, ∠A ≅ ∠DEA by the Isosceles C
Triangle Theorem.
congruent sides.

Got It? 1. a. Is ∠WVS congruent to ∠S? Is TR congruent to TS? Explain. T

b. Reasoning  Can you conclude that △RUV is isoshcsemles1?1gmse_040U5_t027W31.ai
Explain.

R VS

An isosceles triangle has a certain type of symmetry about a line through its vertex

angle. The theorems in this lesson suggest this symmetry, which you will study hinsm11gmse_0405_t027

greater detail in Lesson 9-4.

Lesson 4-5  Isosceles and Equilateral Triangles 251

Theorem 4-5

Theorem If . . . C Then . . . C
If a line bisects the vertex CD # AB and AD B
angle of an isosceles AC ≅ BC and AD ≅ BD
triangle, then the line is ∠ACD ≅ ∠BCD
also the perpendicular
bisector of the base. AD B

You will prove Theorem 4-5 in Exercise 26.

hsm11gmse_0405_t02734.ahism11gmse_0405_t02733

Problem 2 Using Algebra

What does the What is the value of x? A

diagram tell you? Since AB ≅ CB, by the Isosceles Triangle Theorem, 54Њ B
Since AB ≅ CB, △ABC ∠A ≅ ∠C. So m∠C = 54. D xЊ
is isosceles. Since
∠ABD ≅ ∠CBD, BD Since BD bisects ∠ABC, you know by Theorem 4-5 C 36
bisects the vertex angle that BD # AC. So m∠BDC = 90. −

of the isosceles triangle.

m∠C + m∠BDC + m∠DBC = 180 Triangle Angle-Sum Theorem 00000 0
11111 1

54 + 90 + x = 180 Substitute. 22222 2
x = 36 Subtract 144
from each side.hsm11gmse_0405453 _453t0453245373435 5435.ai
66666 6

77777 7

Got It? 2. Suppose m∠A = 27. What is the value of x? 88888 8
99999 9

A corollary is a theorem that can be proved easily using another theorem. Since a hsm11gmse_0405_t05012

corollary is a theorem, you can use it as a reason in a proof.

Corollary to Theorem 4-3

Corollary If . . . Y Then . . . Y
If a triangle is equilateral, XY ≅ YZ ≅ ZX ∠X ≅ ∠Y ≅ ∠Z
then the triangle is
equiangular. XZ XZ

Corollary to Theorem 4-4

Corollary If . . . ∠Y ≅ ∠Zhsm11gYmse_040TX5hY_et≅0n2.Y7.Z3. 6≅.aZiX hsmY11gmse_0405_t02737.
If a triangle is ∠X ≅
equiangular, then the
triangle is equilateral. XZ XZ

252 Chapter 4  Congruent Triangles hsm11gmse_0405_t02737.ahism11gmse_0405_t02736.a

Problem 3 Finding Angle Measures DC

Design  What are the measures of jA, jB, and
jADC in the photo at the right?

A E B

The triangles are equilateral, Let a = measure of one angle.
so they are also equiangular. 3a = 180
Find the measure of each a = 60
angle of an equilateral triangle.
m∠A = m∠B = 60
∠A and ∠B are both angles
in an equilateral triangle.

Use the Angle Addition m∠ADC = m∠ADE + m∠CDE
Postulate to find the
measure of ∠ADC. m∠ADC = 60 + 60
m∠ADC = 120
Both ∠ADE and ∠CDE are
angles in an equilateral
triangle. So m∠ADE = 60
and m∠CDE = 60. Substitute
into the above equation
and simplify.

Got It? 3. Suppose the triangles in Problem 3 are isosceles triangles, where ∠ADE,
∠DEC, and ∠ECB are vertex angles. If the vertex angles each have a
measure of 58, what are m∠A and m∠BCD?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. What is m∠A? 4. What is the relationship between sides and angles for

a. A B b. B each type of triangle?
A 53Њ C a. isosceles
70Њ b. equilateral
C
5. Error Analysis  Claudia drew an isosceles triangle.

2. What is the value of x? She asked Sue to mark it. Explain why the marking of
the diagram is incorrect.

a. hsmE113g0Њmse_0405_bt0. 2Lh73sm8.4a12i1Њ gOxmЊ se_0M405_t02739.ai B

N A
C
xЊ F
D

3. The measure of one base angle of an isosceles triangle

is 23. What are the measures ofhthsme o1th1egrmtwsoe_an0g4l0es5?_t02741.ai

hsm11gmse_0405_t02740.ai

Lesson 4-5  Isosceles anhdsmEq1u1ilgatmersael T_r0ia4n0g5le_st 02742.ai 253

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Complete each statement. Explain why it is true. See Problem 1.

6. VT ≅ ? V

7. UT ≅ ? ≅ YX UY
8. VU ≅ ?

9. ∠VYU ≅ ? TW X

Algebra  Find the values of x and y. See Problem 2.

10. 11. 12. xЊ
yЊ
xЊ 100Њ xЊ hsm4 11gmse_0405_t02743.ai
50Њ yЊ
52Њ y 110Њ

13. An equilateral triangle and an isosceles triangle C See Problem 3.
share a common side. What is the measure of ∠ABC?
B hsm11gmse_0405_t02746.ai

B Apply STEM 14. Ahrcshmit1e1ctgumres  eE_ac0h4f0a5ce_to0f2th7e4G4r.aeai t hsm11gmse_0405_t02745.ai

Pyramid at
66Њ D
Giza is an isosceles triangle with a 76° vertex angle.
A
What are the measures of the base angles?

15. Reasoning  What are the measures of the base angles of a right isosceles
triangle? Explain.

Given isosceles △JKL with base JL, find each value. hsm11gmse_0405_t02747.ai

16. If m∠L = 58, then m∠LKJ = ? . K
17. If JL = 5, then ML = ? .

18. If m∠JKM = 48, then m∠J = ? .

19. If m∠J = 55, then m∠JKM = ? .

J ML

20. Think About a Plan  A triangle has angle measures x + 15, 3x - 35, and 4x. What

type of triangle is it? Be as specific as possible. Justify your answer.

• What do you nkneeodwtaobkonuotwthteosculamssoiffythaetraiannghglelsemm? e1a1sugrmessoef_a0t4ri0an5g_lte0?2748.ai
• What do you

• What type of triangle has no congruent angles? Two congruent angles? Three

congruent angles?

21. Reasoning  An exterior angle of an isosceles triangle has measure 100. Find two
possible sets of measures for the angles of the triangle.

254 Chapter 4  Congruent Triangles

22. Developing Proof  Here is another way to prove the Isosceles Triangle Theorem.
Supply the missing information.

Begin with isosceles △HKJ with KH ≅ KJ . K
Draw a. ? , a bisector of the base HJ .

Given:  KH ≅ KJ , b. ? bisects HJ
Prove:  ∠H ≅ ∠J

Statements Reasons HM J

1) KM bisects HJ . 1) c. ? hsm11gmse_0405_t02749.ai
2) HM ≅ JM 2) d. ?
3) KH ≅ KJ 3) Given
4) KM ≅ KM 4) e. ?
5) △KHM ≅ △KJM 5) f. ?
6) ∠H ≅ ∠J 6) g. ?

23. Supply the missing information in this statement of the Converse of the Isosceles

Proof Triangle Theorem. Then write a proof. R

Begin with △PRQ with ∠P ≅ ∠Q.
Draw a. ? , the bisector of ∠PRQ.

Given:  ∠P ≅ ∠Q, b. ? bisects ∠PRQ

Prove:  PR ≅ QR PS Q

24. Writing  Explain how the corollaries to the Isosceles Triangle Theorem and its

converse follow from the theorems. E

25. Given:  AE ≅ DE, AB ≅ DC hsm11gmse_0405_t02750.ai
Proo f Prove:  △ABE ≅ △DCE

AB CD

26. Prove Theorem 4-5. Use the diagram next to it on page 252.

Proof

STEM 27. a. Communications  In the diagram at the right,

cwahbaltestyopfethoef tsraiamneglheeiisgfhotramneddthbheystgmhroe1u1ngdm? se_01400005_ftt02751.ai Radio
tower
b. What are the two different base lengths 1009 ft tall
of the triangles?
800 ft

c. How is the tower related to each of 600 ft
the triangles?

28. Algebra  The length of the base of an isosceles Cables
triangle is x. The length of a leg is 2x - 5. The
400 ft

perimeter of the triangle is 20. Find x. 200 ft
29. Constructions  Construct equilateral triangle

ABC. Justify your method. 0 ft

450 ft 550 ft

HSM11GMSE_0405_a02297 255
Lesson 4-5  Isosceles and Equilater3ardl Tpraiassng12le-2s 2 -08

Durke

Algebra Find the values of m and n.

30. 31. 126Њ nЊ 32.
nЊ
50Њ mЊ
mЊ mЊ
nЊ

C Challenge CthoeotrhdiirndavteerGteexoomfeatnryis  oFsocreeleaschrigphatirtroifapnhogsilnme.ts1F,i1tnhgdemtrheseaecr_eo0os4irx0dp5ino_aitnt0et2ss 7othf5ae3ta.cacohi uphldosibmnet.11gmse_0405_t02754

33. (4h,s0m) a1n1dg(m0, s4e) _0405_t027523.a4i. (0, 0) and (5, 5) 35. (2, 3) and (5, 6)

36. Reasoning  What measures are possible for the base angles of an acute
isosceles triangle?

Standardized Test Prep

SAT/ACT 37. In isosceles △ABC, the vertex angle is ∠A. What can you prove?

AB = CB m∠B = m∠C ∠A ≅ ∠B BC ≅ AC

38. △LMN ≅ △PQR. What is LM? L Q
x
3 8 10 R P
4 10
2x ϩ 4
N
Short M 12
Response
39. What is the exact area of the base of a circular swimming pool with diameter 16 ft?

1018.29 ft2 1018.3 ft2 64p ft2 256p ft2

40. wSuhpapt oeslsee△doAByoCuannede△d tDoEkFnoawretnoopnrroigveht△trAiaBnCgl≅es.△IfD∠hEBsFm?≅E1x1∠pgElmaaisnne.d_0A4B0≅5_Dt1E1,001.ai

Mixed Review

41. m∠R = 59, m∠T = 93 = m∠H, m∠V = 28, T V G See Lesson 4-4.
See Lesson 2-1.
and RT = GH. What, if anything, can you CH See Lesson 4-2.

conclude about RC and GV? Explain. R

42. Find the pattern of the sequence M, T, W, T, F, . . . Then find the next two terms.

Get Ready!  To prepare for Lesson 4-6, do Exercises 43 and 44.
hsm11gmse_0405_t02755

Can you conclude that the two triangles are congruent? Explain.

43. 44.

256 Chapter 4  Congruent Triangles

Algebra Systems of Linear CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Review Equations
Reviews AM-ARFESI..C9.162  .SAo-lRveEIs.y3s.t6e mSsoolvfelisnyesatermeqsuoaftiloinnesar
Use With Lesson 4-6 eqxaucattliyonans dexaapcptrlyoxainmdaateplpyr(oex.gim.,awteitlhy (ger.agp.,hws)i,thfogcruaspinhgs),
fooncupsaiinrsgoofnlipnaeiarsr eoqf ulianteioanr seqinuatwtionvsairniatbwleos.variables.

You can solve a system of equations in two variables by using substitution.

Example 1

Algebra  Solve the system. y = 3x + 5 y

y = x + 1 2 yϭxϩ1
x
y = x + 1 Start with one equation.
3x + 5 = x + 1 Substitute 3x + 5 for y. O 24
Ϫ2
2x = - 4 Solve for x. y ϭϪ34x ϩ 5
x = -2
Substitute -2 for x in either equation and solve for y.

y = x + 1 = ( -2) + 1 = -1

Since x = -2 and y = 1, the solution is (-2, -1). This is the point of intersection of

the two lines. hsm11gmse_04fb_t05106.ai

The graph of a linear system with infinitely many solutions is one line, and the graph of
a linear system with no solution is two parallel lines.

Example 2

Algebra  Solve the system. x+y=3 y
xϩyϭ3
4x + 4y = 8

x + y = 3 Start with one equation. Ϫ4 Ϫ2 O x
x = 3 - y Solve the equation for x. Ϫ2 24

4(3 - y) + 4y = 8 Substitute 3 - y for x in the second equation. Ϫ4 4x ϩ 4y ϭ 8
12 - 4y + 4y = 8 Solve for y.

12 = 8 False!

Since 12 = 8 is a false statement, the system has no solution.

Exercises hsm11gmse_04fb_t05107.ai

Solve each system of equations.

1. y = x - 4 2. 2x - y = 8 3. 3x + y = 4 4. 2x - 3 = y + 3

y = 3x + 2 x + 2y = 9 -6x - 2y = 12 2x + y = -3

5. y = x + 1 6. x - y = 4 7. y = -x + 2 8. y = 2x - 1
x = y - 1 3x - 3y = 6 2y = 4 - 2x y = 3x - 7

Algebra Review  Systems of Linear Equations 257

4-6 Congruence in Right CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Triangles
MG-ASFRST.9B1.52 .GUs-SeRcoT.n2g.r5u eUnsce c.o. n. gcriuternicaeto. .s.oclrvieteria to
sporolvbelepmrosbalenmd spraonvdeprreolavteiorneslahtiiposnisnhigpesoimn egteroicmfiegturirces.
fMigPur1e,s.MP 3
MP 1, MP 3

Objective To prove right triangles congruent using the Hypotenuse-Leg Theorem

B

One of the tent flaps was
damaged in a storm. Can you use
the other flap as a pattern to
replace it? Explain.

What does the ADC
large triangle tell
you about angles in
the figure?

MATHEMATICAL

PRACTICES

In the diagram below, two sides and a nonincluded angle of one triangle are congruent
to two sides and the nonincluded angle of another triangle.

Q

Lesson B 4 C P 45Њ 5 4
A 45Њ 5 R
Vocabulary
• hypotenuse
• legs of a right

triangle

Notice that the triangles are not congruent. So, you can conclude that Side-Side-Angle

is not a valid method for proving two triangles congruent. This method, however, works

in thehsspmec1ia1lgcmaseseo_f 0ri4gh0t6t_rita0n2g4le8s,6w.ahiere hthsemri1gh1tgamngslee_s 0ar4e0t6h_etn0o2n4in8c7lu.adied angles.
In a right triangle, the side opposite the
right angle is called the hypotenuse. It is The right angle Leg Hypotenuse
the longest side in the triangle. The other always “points“ to Leg
two sides are called legs. the hypotenuse.

Essential Understanding  You can prove that two triangles are congruent

without having to show that all corresponding parts are congruent. In this lesson,

you will prove right triangles congruent by using one pair of right angles, a pair of
hsm11gmse_0406_t02496.ai
hypotenuses, and a pair of legs.

258 Chapter 4  Congruent Triangles

Theorem 4-6  Hypotenuse-Leg (HL) Theorem

Theorem If . . . Then . . .
If the hypotenuse △PQR and △XYZ are right △s , △PQR ≅ △XYZ
and a leg of one right PR ≅ XZ, and PQ ≅ XY
triangle are congruent
to the hypotenuse PX
and a leg of another
right triangle, then Q RY Z
the triangles are
congruent.

To prove the HL Theorem youhwsimll n1e1egdmtosder_a0w4a0u6x_iltia0r2y4li9n9e.satio make a third triangle.

Proof Proof of Theorem 4-6: Hypotenuse-Leg Theorem P X Z
Q
Given: △PQR and △XYZ are right triangles, with right angles RY
Q and Y. PR ≅ XZ and PQ ≅ XY . X

Prove: △PQR ≅ △XYZ
Proof: On △XYZ, draw ZY>.

Mark point S so that YS = QR. Then, △PQR ≅ △XYS by SAS. hsm11gmse_0406_t05108

Since corresponding parts of congruent triangles are congruent, S Y Z
PR ≅ XS. It is given that PR ≅ XZ, so XS ≅ XZ by the Transitive

Property of Congruence. By the Isosceles Triangle Theorem,

∠S ≅ ∠Z, so △XYS ≅ △XYZ by AAS. Therefore, △PQR ≅ △XYZ

by the Transitive Property of Congruence. hsm11gmse_0406_t02504

Key Concept  Conditions for HL Theorem

To use the HL Theorem, the triangles must meet three conditions.

Conditions
• There are two right triangles.
• The triangles have congruent hypotenuses.
• There is one pair of congruent legs.

Lesson 4-6  Congruence in Right Triangles 259

Proof Problem 1 Using the HL Theorem

On the basketball backboard brackets shown below, ∠ADC and ∠BDC are
right angles and AC ≅ BC. Are △ADC and △BDC congruent? Explain.

How can you A
visualize the two DC
right triangles? B
Imagine cutting △ABC
along DC. On either

side of the cut, you get

triangles with the same

leg DC.

• You are given that ∠ADC and ∠BDC are right angles.
So, △ADC and △BDC are right triangles.

• The hypotenuses of the two right triangles are AC and BC.
You are given that AC ≅ BC.

• DC is a common leg of both △ADC and △BDC. DC ≅ DC by the Reflexive Property
of Congruence.

Yes, △ADC ≅ △BDC by the HL Theorem. PQ

Got It? 1. a. Given:  ∠PRS and ∠RPQ are
right angles, SP ≅ QR

Prove:  △PRS ≅ △RPQ SR

b. Reasoning  Your friend says, “Suppose you have two right triangles with

congruent hypotenuses and one pair of congruent legs. It does not matter

which leg in the first triangle is congruent to which leg in the second

triangle. The triangles will be congruent.” Is yhosumr f1ri1egndmcsoer_re0c4t?0E6x_ptl0ai2n5. 44

260 Chapter 4  Congruent Triangles

Proof Problem 2 Writing a Proof Using the HL Theorem B D
E
How can you get Given: BE bisects AD at C,
started?
Identify the hypotenuse AB # BC, DE # EC, AB ≅ DE C
of each right triangle.
Prove that the Prove: △ABC ≅ △DEC A
hypotenuses are
congruent. BE bisects AD. AC ≅ DC
Given Def. of bisector

AB ⊥ BC ∠ABC and △ABC and △DEC hsm11gmse_0406_t02545
DE ⊥ EC ∠DEC are are right .
right ⦞. Def. of right triangle △ABC ≅ △DEC
Given Def. of ⊥ lines HL Theorem

AB ≅ DE A C
Given
BD
Got It? 2. Given:  CD ≅ EA, AD is the perpendicular E
bisector of CE

hsm11g mPsroev_e0:4  △06C_BtD02≅54△6E.aBiA

Lesson Check hsm11gmse_0406_t02548.ai

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Are the two triangles congruent? If so, write the 5. Vocabulary  A right triangle has side lengths of
congruence statement. 5 cm, 12 cm, and 13 cm. What is the length of the
hypotenuse? How do you know?
1. A 10 2. L N
10 F D M 6. Compare and Contrast  How do the HL Theorem and
the SAS Postulate compare? How are they different?
6 Explain.

C 6B E P O 7. Error Analysis  Your classmate says that there is not
enough information to determine whether the two

3. B 4. R triangles below are congruent. Is your classmate

QR correct? Explain.

hsm11gmse_0406_t02604.aihsm11gmse_0406_t02605.ai M J

K XT

LT V L
K

hsm11gmse_0406_t02606.aihsm11gmse_0406_t02607.ai hsm11gmse_0406_t02549.ai

Lesson 4-6  Congruence in Right Triangles 261

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice 8. Developing Proof  Complete the flow proof. S R T See Problem 1.
P
Given:  PS ≅ PT , ∠PRS ≅ ∠PRT △PRS and △PRT
Prove:  △PRS ≅ △PRT are right .

∠PRS and ∠PRT are ≅. ∠PRS and ∠PRT b.
Given are right ⦞.

a.

∠PRS and ∠PRT PS ≅hPsTm11gmse_0406_t02550.ai
are supplementary.
⦞ that form a linear c. △PRS ≅ △PRT
pair are supplementary.
PR ≅ PR e.

d.

9. Developing Proof  Complete the paragraph proof. B D
A E
Given:  ∠A and ∠D are right angles, AB ≅ DE

Prove: h△smAB1E1g≅m△sDe_E0B406_t02551.ai

Proof:  It is given that ∠A and ∠D are right angles. So, a. ? by the
definition of right triangles. b. ? , because of the Reflexive Property of
Congruence. It is also given that c. ? . So, △ABE ≅ △DEB by d. ? .

10. Given:  HV # GT , GH ≅ TV , 11. Given:  PM ≅ RJ , hsm11gmsSee_e0P4r0o6b_letm0225.52.ai
Proof I is the midpoint of HV Proof
PT # TJ , RM # TJ ,

Prove:  △IGH ≅ △ITV M is the midpoint of TJ

G Prove:  △PTM ≅ △RMJ

I V P

H TM J
T

R

B Apply Algebra  For what values of x and y are the triangles congruent by HL?

12. hsm11gmse_0406_t02553.ai 13. 3y ϩ x

x x ϩ 3 3y y ϩ 1 hsm11gmy Ϫsex_0406_t0255x 4ϩ 5

yϩ5

14. Study Exercise 8. Can you prove that △PRS ≅ △PRT without using the HL
Theorem? Explain.

hsm11gmse_0406_t02555 hsm11gmse_0406_t02556

262 Chapter 4  Congruent Triangles

15. Think About a Plan  △ABC and △PQR are C
right triangular sections of a fire escape,
as shown. Is each story of the building the AB
same height? Explain. R

• What can you tell from the diagram?
• How can you use congruent triangles here?

16. Writing  “A HA!” exclaims your classmate.
“There must be an HA Theorem, sort of
like the HL Theorem!” Is your classmate
correct? Explain.

PQ

17. Given:  RS ≅ TU , RS # ST , TU # UV , 18. Given:  △LNP is isosceles with base NP,
Proof MN # NL, QP # PL, ML ≅ QL
Proof T is the midpoint of RV

Prove:  △RST ≅ △TUV Prove:  △MNL ≅ △QPL

R L

ST

QU V MQ
NP

Constructions  Copy the triangle and construct a triangle congruent
to it using the given method.

19. SAS hsm11gmse_0406_t02557 20. HL hsm11gmse_0406_t02558

21. ASA 22. SSS

23. Given:  △GKE is isosceles with base GE, 24. Given:  LO bisects ∠MLN ,
Proof
Proof ∠L and ∠D are right angles, and OM # LM, ON # LN

K is the midpoint of LD . Prove:  △LMO ≅ △LNOhsm11gmse_0406_t02559

Prove:  LG ≅ DE M
L K DO

GE LN
25. Reasoning  Are the triangles congruent? Explain.
CF
hsm11gmse_0406_t02560
hsm1131gmse_50B406E_5t02561
D
A 13

Lesson 4-6  Congruence in Right Triangles 263

26. a. Coordinate Geometry  Graph the points A( -5, 6), B(1, 3), D( -8, 0), and E( -2, -3).

Draw AB, AE, BD, and DE. Label point C, the intersection of AE and BD.

b. Find the slopes of AE and fBoDr <A. HE>oawndwo<BuDl>d. you describe ∠ACB and ∠ECD?
c. Algebra  Write equations What are the coordinates of C?

d. Use the Distance Formula to find AB, BC, DC, and DE.

e. Write a paragraph to prove that △ABC ≅ △EDC.

C Challenge Geometry in 3 Dimensions  For Exercises 27 and 28, use the figure at the right. B

27. Given:  BE # EA, BE # EC, △ABC is equilateral
Proo f Prove:  △AEB ≅ △CEB

28. Given:  △AEB ≅ △CEB, BE # EA, BE # EC AE C
Can you prove that △ABC is equilateral? Explain.

Standardized Test Prep School (S)
Park (P)
SAT/ACT 29. You often walk your dog around the neighborhood. Café (C) hsm11gmse_0406_t02562
Based on the diagram at the right, which of the Library (L)
following statements about distances is true? Home (H)

SH = LH SH 7 LH

PH = CH PH 6 CH

30. What is the midpoint of LM with endpoints L(2, 7) and M(5, -1)?

(3.5, 3) (2, 4.5) X

(3.5, 4) (7, 6)
Short 31. In equilateral △XYZ, name four pairs of congruent right hsm11gmseP_0406_Qt02563
Response S
triangles. Explain why they are congruent.

YRZ

Mixed Review
For Exercises 32 and 33, what type of triangle must △STU be? Explain. hsm11gmse_04S0e6e_Lt0es2s5o6n44-5.

32. △STU ≅ △UTS 33. △STU ≅ △UST

G et Ready!  To prepare for Lesson 4-7, do Exercises 34–36.

Can you conclude that the triangles are congruent? Explain. See Lessons 4-3 and 4-6.
36. △RST and △ABC
34. △ABC and △LMN 35. △LMN and △HJK

B T L J

AC S K H
RN M

hsm11gmse_0406_t02565 hsm11gmse_0406_t025h6s7m11gmse_0406_t02568

264 Chapter 4  Congruent Triangles