cgpe-0502 07/24/2001 8:02 PM Page 240 Activity 5.2 Congruent...

240 Chapter 5 Congruent Triangles

Activity Congruent TrianglesCongruent TrianglesCongruent Triangles5.25.2

Question

Can you determine that two triangles are congruent without listing

all of the corresponding congruent sides and angles?

Think About It

1. What do you notice about the triangles you made?

2. Based on this activity, if you know that two sides of a triangle are congruent to two sides of another triangle, what other information do you need to know to show that the triangles are congruent?

LOOK BACK

For help with using aprotractor, see p. 36.

Student Help

• two pencils• protractor• straightedge

Materials

Explore

�1 On a piece of paper, placetwo pencils so their erasersare at the center of aprotractor. Arrange thepencils to form a 45� angle.

�2 Mark two vertices of atriangle by pressing thepencil points to the paper.Mark the center of theprotractor as the third vertex.

�3 Remove the pencils andprotractor and draw the sides of the triangle.

�4 Repeat Steps 1–3 using the same pencils. Try to make a triangle that has a 45� angle but is not congruent to the one you made.

cgpe-0502 07/24/2001 8:02 PM Page 240

5.2 Proving Triangles are Congruent: SSS and SAS 241

Goal

Show triangles arecongruent using SSS and SAS.

Key Words

• proof

The Geo-Activity suggests the following postulate.

5.25.2 Proving Triangles areCongruent: SSS and SAS

�2 Open your compass to measureAB&* of TABC. Use this length toconstruct DE&* so that it iscongruent to AB&*.

�4 Open your compass to measureBC&*. Use this length to draw anarc centered at E that intersectsthe arc from Step 3.

�3 Open your compass to measureAC&*. Use this length to draw anarc with the point of thecompass at D.

�5 Label the point of intersection F.Then draw TDEF. Measure theangles of the triangles toconfirm that TABC c TDEF.

D E

F

D E

D E

F

D E

Copying a TriangleCopying a TriangleCopying a TriangleGeo-Activity

�1 Use a straightedge to draw a large triangle. Label it TABC.

Side-Side-Side Congruence Postulate (SSS)

Words If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent.

Symbols If Side MN&** c QR&*, and

Side NP&* c RS&*, and

Side PM&** c SQ&*,

then TMNP c TQRS.

POSTULATE 12

N

M

P R S

P

A B

C

cgpe-0502 07/24/2001 8:02 PM 41


Does the diagram give enough information to show that the triangles are congruent? Explain.

Solution

From the diagram you know that HJ&* c LJ&* and HK&** c LK&*.

By the Reflexive Property, you know that JK&* c JK&*.

ANSWER � Yes, enough information is given. Because correspondingsides are congruent, you can use the SSS CongruencePostulate to conclude that THJKcTLJK.

EXAMPLE 111 Use the SSS Congruence Postulate

Does the diagram give enough information to use the SAS CongruencePostulate? Explain your reasoning.

a. b.

Solution

a. From the diagram, you know that AB&* c CB&* and DB&* c DB&*.

The angle included between AB&* and DB&* is aABD.

The angle included between CB&* and DB&* is aCBD.

Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that TABDcTCBD.

b. You know that GF&* c GH&** and GE&* c GE&*. However, the congruentangles are not included between the congruent sides, so you cannot use the SAS Congruence Postulate.

F

EG

H

B CA

D

EXAMPLE 222 Use the SAS Congruence Postulate

In the triangle below,aB is the includedangle between sidesAB&* and BC&*.

Visualize It!

H

K

JL

A B

C

included angle

Side-Angle-Side Congruence Postulate (SAS)

Words If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Symbols If Side PQ&* c WX&&, and

Angle aQ c aX, and

Side QR&* c XY&**,

then TPQR c TWXY.

POSTULATE 13

X

W YW

P

P R

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Writing Proofs A is a convincing argument that shows why astatement is true. A two-column proof has numbered statements andreasons that show the logical order of the argument. Each statementhas a reason listed to its right.

proof

• List the given information first.

• Use information from the diagram.

• Give a reason for every statement.

• Use given information, definitions, postulates, and theorems as reasons.

• List statements in order. If a statement relies on another statement,list it later than the statement it relies on.

• End the proof with the statement you are trying to prove.

HOW TO WRITE A PROOF

Write a two-column proof that shows T JKLcTNML.

Given � JL&* c NL&*L is the midpoint of KM&**.

Prove � T JKLcTNML

Solution

The proof can be set up in two columns. The proof begins with the given information and ends with the statement you are trying to prove.

Statements Reasons

S 1. JL&* c NL&* 1. Given

2. L is the midpoint 2. Givenof KM&**.

A 3. aJLKc aNLM 3. Vertical AnglesTheorem

S 4. KL&* cML&** 4. Definition ofmidpoint

5. TJKLcTNML 5. SAS Congruence Postulate

EXAMPLE 333 Write a Proof

M

N

J

LK

These are the givenstatements.

This information is from the diagram.

Statement 4 follows fromStatement 2.

Statement 5 follows fromthe congruences ofStatements 1, 3, and 4.

STUDY TIP

You can remind yourselfabout side and anglecongruences by writingletters as shown.

Student Help

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You are making a model of the window shown in the photo. You know that DR&* ∏ AG&* and RA&* c RG&*. Write a proof to show that TDRAcTDRG.

Solution

�1 Make a diagram and label it with the given information.

�2 Write the given information and the statement you need to prove.

Given � DR&* ∏ AG&*, RA&* c RG&*

Prove � TDRAcTDRG

�3 Write a two-column proof. List the given statements first.

Statements Reasons

1. RA&* c RG&* 1. Given

2. DR&* ∏ AG&* 2. Given

3. aDRA and aDRG are 3. ∏ lines form right angles.right angles.

4. aDRAc aDRG 4. Right angles are congruent.

5. DR&* c DR&* 5. Reflexive Property of Congruence

6. TDRAcTDRG 6. SAS Congruence Postulate


EXAMPLE 444 Prove Triangles are Congruent

Prove Triangles are Congruent

1. Fill in the missing statements and reasons.

Given � CB&* c CE&*, AC&* c DC&**

Prove � TBCAcTECD

Statements Reasons

1. CB&* c CE&* 1. _________?_________

2. _________?_________ 2. Given

3. aBCAc aECD 3. _________?_________

4. TBCAcTECD 4. _________?_________

D

A GR

DD

AA GGRR

B

A E

DC

STUDY TIP

Think about what youcan say given that DR&* ∏ AG&*.

Then think about what other information youcan deduce from thediagram.

Student Help

MORE EXAMPLES

More examples atclasszone.com

IStudent HelpI C L A S S Z O N E . C O M

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Use TJKL to name the angle included between the two sides.

1. JK&* and KM&**

2. JK&* and MJ&*

3. KL&* and JL&

4. KM&** and LM&**

5. LK&* and KM&**

Decide whether enough information is given to show that the

triangles are congruent. If so, tell which congruence postulate

you would use.

6. TABC, TDEC 7. TFGH, TJKH 8. TPQR, TSRQ

Naming Included Angles Use the diagram shown to name the angle

included between the two sides.

9. AB&* and BD&* 10. CD&* and BC&*

11. AC&* and CB&* 12. BA&* and AD&*

13. DC&** and BD&* 14. BD&* and BC&*

Using SSS Decide whether enough information is given to use

the SSS Congruence Postulate. Explain your reasoning.

15. 16. 17.

P RS

P

J

K

M

L

A DB

C E

B

A CD

Practice and Applications

R

P P

S

F

H

J

G

K

A B

E D

C

Skill Check

Vocabulary Check

Guided Practice

Exercises5.25.2

Extra Practice

See p. 683.

Example 1: Exs. 15–17,21–31

Example 2: Exs. 18–31Example 3: Exs. 34–36Example 4: Exs. 34–36

Homework Help

K

J LM

cgpe-0502 2/6/02 10:39 AM Page 245


Using SAS Congruence Decide whether enough information is given

to use the SAS Congruence Postulate. Explain your reasoning.

18. 19. 20.

Decide whether enough information is given

to show that the triangles are congruent. If so, tell which congruence

postulate you would use.

21. 22. 23.

24. 25. 26.

Textiles In Exercises 27 and 28, use the photo of the Navajo rug. In

the triangles on the rug, BC&* c DE&* and AC&* c CE&*.

27. What additional information is needed to use the SSS CongruencePostulate to show that TABCcTCDE ?

28. What additional information is needed to use the SAS CongruencePostulate to show that TABCcTCDE ?

Missing Information Determine what single piece of information

you need to know in order to use either the SSS or SAS Congruence

Postulate to show that the triangles are congruent.

29. 30. 31.

R

S

P

P

L M

J K

D E

FA B

C

D

A B C

R S T

U

X

W Y

Z

M N

PP

G H

JK

B

C

D

A

E

F

You be the Judge

L

J K

M

N

A

B C

E

D

R

P

S P

cgpe-0502 07/24/2001 8:02 PM Page 246

Quilting Use the labels on the diagrams to explain how you know

that the triangles are congruent.

32. WXYZ is a square. 33. AB&* � CD&*

Reasoning In Exercises 34 and 35, fill in the missing statements

and reasons.

34. Given � EF&* c GH&*FG&* c HE&**

Prove � TEFGcTGHE

35. Given � SP&* c TP&*PQ&* bisects aSPT.

Prove � TSPQcTTPQ

Statements Reasons

1. SP&* c TP&* 1. Given

2. PQ&* bisects aSPT. 2. _________?_________

3. aSPQc aTPQ 3. _________?_________

4. _________?_________ 4. Reflexive Prop. of Cong.

5. TSPQ cTTPQ 5. _________?_________

P

P

TS

Statements Reasons

1. EF&* c GH&* 1. Given

2. FG&* c HE&** 2. _________?_________

3. GE&* c GE&* 3. _________?_________

4. TEFGcTGHE 4. _________?_________

G F

H E


TRIANGLES To create thequilt pattern in Ex. 33, squaresare cut out of two rectangularstrips of fabrics sewn together.

Application LinksC L A S S Z O N E . C O M

Quilting

HOMEWORK HELP

Extra help with problemsolving in Exs. 34–35 is at classzone.com

IStudent HelpI C L A S S Z O N E . C O M

W

X

Z

Y D

A

C

B

cgpe-0502 07/24/2001 8:02 PM Page 247

36. Reasoning Fill in the missing statements and reasons.

Given � AC&* c BC&*M is the midpoint of AB&*.

Prove � TACMcTBCM

37. Error Analysis Using the diagram below, Maria was asked whetherit can be shown that TABC cTDEF. Explain her error.

Challenge Write a proof to show that the triangles are congruent.

38. TABD and TCBD 39. XZ&* c YZ&*, and ZM&**are equilateral. bisects aYZX.

40. Multiple Choice In TRST and TABC, RS&* c AB&*, ST&* c BC&*, andTR&* c CA&*. Which angle is congruent to aT ?

�A aA �B aR �C aC �D aB

41. Multiple Choice In the diagram below, TDEF is equilateral and G is the midpoint of DE&*. Which of the statements is not true?

�F DF&* c EF&* �G DG&** c DF&*

�H DG&** c EG&* �J TDFGcTEFG

Standardized Test

Practice

M

Z

Y

X

A C

B

D

Statements Reasons

S 1. AC&* c BC&* 1. Given

2. M is the midpoint of AB&*. 2. _________?_________

S 3. _________?_________ 3. Definition of midpoint

S 4. CM&** c CM&** 4. _________?_________

5. TACMcTBCM 5. _________?_________

C

A BM


D E

F

G

A B

E D

C

F

Maria

STUDY TIP

It is helpful to labelcongruent sides andcongruent angles insteps of a proof.

Student Help

TABC c TDEF by the SAS CongruencePostulate.

cgpe-0502 07/24/2001 8:02 PM Page 248

Classifying Angles Use the diagram to determine whether the

angles are corresponding, alternate interior, alternate exterior,

or same-side interior angles. (Lesson 3.3)

42. a1 and a5 43. a2 and a6

44. a7 and a2 45. a5 and a8

46. a9 and a4 47. a5 and a9

Using the Triangle Inequality Determine whether it is possible to

draw a triangle with the given side lengths. (Lesson 4.7)

48. 14, 8, 25 49. 20, 10, 28 50. 16, 14, 30

Evaluating Square Roots Evaluate. Give the exact value if possible. If

not, approximate to the nearest tenth. (Skills Review, p. 669)

51. �3� 52. �1�2� 53. �4�0� 54. �1�5�9�

55. �1�4�.7�6� 56. �0�.8�7� 57. �1�.1�2� 58. �4�0�.8�5�

In the diagram, TABC c TQPR. Complete the statement with the

corresponding congruent part. (Lesson 5.1)

1. aRc __?__ 2. AB&* c __?__

3. TBACc __?__ 4. TRPQc __?__

5. Write a congruence statement for the congruent triangles shown at the right. (Lesson 5.1)

Decide whether enough information is given to show that the

triangles are congruent. If so, tell which congruence postulate

you would use. Explain your reasoning. (Lesson 5.2)

6. 7. 8.

K

F

J NM

G

F

D E

CB

XY

A

Z

G

X

E

YZ

F

Quiz 1

Algebra Skills

Mixed Review


12

76

543

98

BA P

C R

P

cgpe-0502 2/6/02 10:40 AM Page 249

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