Using Congruent Triangles: 4-4 CPCTC

[Pages:5]4-4

Using Congruent Triangles:

CPCTC

What You'll Learn

? To use triangle congruence

and CPCTC to prove that parts of two triangles are congruent

. . . And Why

To measure distance indirectly, as in Example 2

Check Skills You'll Need

GO for Help Lesson 4-1

In the diagram, kJRC O k HVG. 1?4. See back of book.

1. List the congruent corresponding angles.

2. List the congruent corresponding sides.

You are given that kTIC O kLOK. 3. List the congruent corresponding angles. J

C H R

G V

4. List the congruent corresponding sides.

New Vocabulary ? CPCTC

1 Proving Parts of Triangles Congruent

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

Proof 1 EXAMPLE Real-World Connection

Real-World Connection

Shapes formed by the ribs, stretchers, and shaft are congruent whether an umbrella is open or closed.

Quick Check

Umbrella Frames In an umbrella frame, the stretchers are congruent and they open to angles of equal measure.

Given: SL > SR, &1 > &2

Prove that the angles formed by the shaft and the ribs are congruent.

Prove: &3 > &4 Proof: It is given that SL > SR and &1 > &2. SC > SC by the Reflexive Property of Congruence. #LSC > #RSC by SAS, so &3 > &4 by CPCTC.

C 3 4 Rib

L

R

5

6

12 Stretcher

S

Shaft

1 Given: &Q > &R,

P

Q

&QPS > &RSP

Prove: SQ > PR

R

S

It is given that lQ O lR and lQPS O lRSP. PS O SP by the

Reflexive Prop. of O. #QPS O #RSP by ASA. SQ O PR by CPCTC.

You can use congruent triangles and CPCTC to measure distances, such as the

distance across a river, indirectly.

Lesson 4-4 Using Congruent Triangles: CPCTC 221

4-4

1. Plan

Objectives

1 To use triangle congruence and CPCTC to prove that parts of two triangles are congruent

Examples

1 Real-World Connection 2 Real-World Connection

Math Background

Proving triangles congruent is usually not an end in itself. When polygons are divided into triangles, applying CPCTC to congruent triangles can lead to congruence statements that otherwise would have been difficult or impossible to prove.

More Math Background: p. 196D

Lesson Planning and Resources

See p. 196E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to: Listing Congruent Parts Lesson 4-1: Example 1 Extra Skills, Word Problems, Proof

Practice, Ch. 4

Special Needs L1 Have students reenact the measurement technique presented in Example 2. Then have students find the actual measure and compare it with their results.

Below Level L2 Use a real umbrella to demonstrate that the corresponding angles in Example 1 are congruent for all openings of the umbrella.

learning style: tactile

learning style: visual

221

2. Teach

Guided Instruction

2 EXAMPLE Tactile Learners

Students can reenact the officer's measurement technique by measuring the distance across a playing field.

Error Prevention!

Remind students that corresponding parts of congruent triangles are clearly indicated by the order of vertices in the triangle congruence statement. Have students practice identifying corresponding parts without referring to diagrams.

PowerPoint

Additional Examples

1 What other congruence statements can you prove from the diagram and paragraph proof in Example 1? lCLS O lCRS, CL O CR

2 The Example 2 Given states that &DEG and &DEF are right angles. What conditions must hold for that to be true? The officer must stand perpendicular to the ground.

Resources

? Daily Notetaking Guide 4-4 L3

? Daily Notetaking Guide 4-4--

Adapted Instruction

L1

Closure

Explain why CPCTC is useful. Sample: It allows you to make conclusions about the other parts of congruent triangles.

Proof 2 EXAMPLE Real-World Connection

Quick Check

History According to legend,

one of Napoleon's officers used

congruent triangles to estimate

the width of a river. On the

riverbank, the officer stood up

F

straight and lowered the visor

of his cap until the farthest thing

D

he could see was the edge of the

opposite bank. He then turned

E

and noted the spot on his side

of the river that was in line with

his eye and the tip of his visor.

G not to scale

Given: &DEG and &DEF are right angles; &EDG > &EDF.

The officer then paced off the distance to this spot and declared that distance to be the width of the river! Use congruent triangles to prove that he was correct.

Prove: EF > EG

Statements

Reasons

1. &EDG > &EDF

2. DE > DE 3. &DEG and &DEF are right angles. 4. &DEG > &DEF 5. #DEF > #DEG 6. EF > EG

1. Given

2. Reflexive Property of Congruence 3. Given 4. All right angles are congruent. 5. ASA Postulate 6. CPCTC

2 About how wide was the river if the officer stepped off 20 paces and each pace was about 212 ft long? 50 ft

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

1. Developing Proof State why the two triangles

LM

Example 1

are congruent. Give the congruence statement. Then list all the other parts of the triangles that

(page 221)

GO

for Help

are congruent by CPCTC. SAS; kKLJ O kOMN; lK O l O, lJ O lN, JK O NO

K

JN

O

Proof 2. Given: &ABD > &CBD,

2. kABD O kCBD by ASA

&BDA > &BDC

because BD O BD by Reflexive Prop. of O;

Prove: AB > CB

AB O CB by CPCTC

B

3. kMOE O kREO by SSS

A

C

because OE O OE by

Reflexive Prop. of O;

lM O lR by CPCTC

D

3. Given: OM > ER, ME > RO

Prove: &M > &R O

R M

E

222 Chapter 4 Congruent Triangles

222

Advanced Learners L4 Have students discuss why DEG and DEF are both right angles in Example 2, and then suggest places on a riverbank where the angles would not be right angles.

learning style: verbal

English Language Learners ELL Some students may be unfamiliar with the expression "paced off." You may want to demonstrate what it means by pacing off the length of the classroom.

learning style: visual

Example 2 (page 222)

B Apply Your Skills

4. Developing Proof Two cars of the same model have hood braces that are identical, connect to the body of the car in the same place, and fit into the same slot in the hood.

Given: CA > VE, AR > EH, RC > HV

C R

A

V H

E

Complete the proof that the hood braces hold the hoods open at the same angle.

Prove: &ARC > &EHV

Proof: It is given that the three sides of the triangles are congruent,

so #ARC > #EHV by a. 9. Thus, &ARC > &EHV by b. 9.

SSS

CPCTC

5. Earth Science Some distances are best measured indirectly.

Sinkhole Swallows House

The large sinkhole in this photo occurred

suddenly in 1981 in Winter Park, Florida,

following a severe drought. Increased

water consumption lowers the water

table. Sinkholes form when caverns in

8. lPKL O lQKL by def. of l bisect, and KL O KL by Reflexive Prop. of O, so the > are O by SAS.

9. KL O KL by Reflexive Prop. of O; PL O LQ by Def. of # bis.; lKLP O lKLQ by Def. of #; the > are O by

the underlying limestone dry up and collapse.

A geometry class indirectly measured the distance across a sinkhole. The distances they measured are shown in the diagram. 26.5 yd Explain how to use their measurements to find the distance across the sinkhole. See margin.

40 yd 30 yd 40 yd 30 yd

SAS.

Proof 6. Given: &SPT > &OPT,

7. Given: YT > YP, &C > &R

10. lKLP O lKLQ because all rt ' are O;

SP > OP 6?7.

&T > &P

KL O KL by Reflexive

Prove: &S > &O See back of book. Prove: CT > RP

Prop. of O; and lPKL O lQKL by def. of

S

T

O

C

R

bisect; the > are O by

ASA.

Y

P

T

P

GO nline

Copy and mark the figure to show the given information.

K

Explain how you would use SSS, SAS, ASA, or AAS

Homework Help

with CPCTC to prove lP O lQ.

Visit: Web Code: aue-0404

8. Given: PK > QK, KL bisects &PKQ. 8?10. See left.

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

10. Given: KL ' PQ, KL bisects &PKQ. Proof 11. Given: &QPS > &RSP, &Q > &R

P

L

Q

P

Prove: PQ > SR See back of book.

Q

R

S

Lesson 4-4 Using Congruent Triangles: CPCTC 223

3. Practice

Assignment Guide

1 A B 1-19 C Challenge

20-21

Test Prep Mixed Review

22-25 26-30

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 3, 4, 12, 13, 15.

Connection to History Exercise 5 The city of Ubar,

believed to have existed in southwestern Oman from 2800 B.C. to about A.D.100, fell into a sinkhole created by the collapse of an underground limestone cavern. Called "Atlantis of the Sands" by Lawrence of Arabia, the fabled city was located in 1992 using images from satellites and spacecraft.

Exercises 13, 17 The constructions in Lesson 1-7 were presented without justification. Discuss as a class how these exercises provide rationales for those constructions.

GPS Guided Problem Solving

L3

Enrichment

L4

Reteaching

L2

Adapted Practice

L1

PNamreactice

Class

Date

L3

Practice 4-4

Using Congruent Triangles: CPCTC

Explain how you can use SSS, SAS, ASA, or AAS with CPCTC to prove each statement true.

1. A C

2. HE FG

3. K P

B

F

E

J L M

A

D

C

G H

N KP

4. QST SQR

Q

R

T

S

5. U W V

U

W

X

6. ZA AC

Y

Z

A

C

B

7. FG DG

D

F

E

G

8. JK KL M

J

K L

H

9. N Q

N

P

R

Q

Write a Plan for Proof.

10. Given: BD # AB, BD # DE, BC CD Prove: A E

A

B

C

D

E

11. Given: FJ GH, JFH GHF Prove: FG JH

F

G

J

H

? Pearson Education, Inc. All rights reserved.

223

4. Assess & Reteach

PowerPoint

Lesson Quiz

1. What does "CPCTC" stand for? Corresponding parts of O triangles are O.

Use the diagram for Exercises 2 and 3.

A

C

B

M

2. Tell how you would show ABM ACM. You are given two pairs of O ls, AM O AM by the Reflexive Prop., so kABM O kACM by ASA.

3. Tell what other parts are congruent by CPCTC. AB O AC, BM O CM, lB O lC

Use the diagram for Exercises 4 and 5.

R

S U

Q

T

4. Tell how you would show RUQ TUS. You are given a pair of O ls and a pair of O sides, lRUQ O lTUS because vert. angles are O, so kRUQ O kTUS by AAS.

5. Tell what other parts are congruent by CPCTC. RQ O TS, UQ O US, lR O lT

5. The > are O by SAS so the distance across the sinkhole is 26.5 yd by CPCTC.

6. lSPT lOPT, SP O OP (Given),

224

12. Yes, kABD O kCBD by SSS so lA O lC by CPCTC.

GO for Help

For a guide to solving Exercise 14, see page 226.

A 12. Writing Karen cut this pattern for the stained

glass shown here so that AB = CB and AD = CD. D

B

Must &A be congruent to &C? Explain. C

13. Constructions The construction of a line perpendicular

GPS to line / through point P on / is shown here.

a. Which lengths or distances are equal by

C

construction? AP O PB; AC O BC* )

b. Explain why you can conclude that CP is perpendicular to /. (Hint: Do the construction. Then draw CA and CB.)

A

P

B

See margin.

14. Error Analysis The proof is incorrect. Find the error and tell how you would

correct the proof. See back of book. B

Given: &A > &C, BD bisects &ABC.

Prove: AB > CB

12

Statements

Reasons

1. &A > &C

1. Given

ADC

2. BD bisects &ABC. 2. Given

15. BA O BC is given; BD O BD by the

Reflexive Prop. of O and since BD bisects lABC, lABD O lCBD by Def. of an l bisector;

3. &1 > &2 4. AD > CD 5. #ABD > #CBD 6. AB > CB

3. Definition of bisect 4. Definition of bisect 5. AAS Theorem 6. CPCTC

thus, kABD O kCBD by SAS; AD O DC by Proof 15. Given: BA > BC,

16. Given: / ' AB, / bisects AB at C,

CPCTC so BD bisects

AC by Def. of a bis.; lADB O lCDB by CPCTC and lADB and lCDB are supp.; thus,

BD bisects &ABC.

Prove: BD ' AC, BD bisects AC. See left.

B

P is on /.

Prove: PA = PB See back of

P

book.

lADB and lCDB are

right ' and BD # AC

by Def. of #.

17. kABX O kACX by SSS,

so lBAX O lCAX) by

CPCTC. Thus AX bisects lBAC by the Def. of l bisector.

A

D

C

A CB

17. Constructions In the construction of the bisector of &A below, AB > AC

because they are radii of the same circle. BX > CX becau) se both arcs had the

same compass setting. Tell why you can conclude that AX bisects &BAC.

Problem Solving Hint

In the third diagram, what two triangles must be congruent, and why?

B

B

B

X

X

A

C

A

C

A

C

Proof 18. Given: BE ' AC, DF ' AC, 18?19. 19. Given: JK 6 QP, JK > QP

BE > DF, AF > EC See back Prove: KQ bisects JP.

Prove: AB > DC

of book. K

B

FC

AE

D

M J

P Q

224 Chapter 4 Congruent Triangles

PT O PT (Reflexive Prop.) kSPT O kOPT (SAS), lS lO (CPCTC) 7. YT O YP, lC O lR, lT O lP (Given), lCYT lRYP (If 2 l or a k

are O to 2 l of another, the 3rd l are O.), kCYT O kRYP (ASA), CT O RP (CPCTC)

13. b. The diagram is constructed in such a way that the > are O

by SSS. lCPA O lCPB by CPCTC.

Since these ' are O and suppl., they are

4 right '. Thus, CP is # to /.

C Challenge Proof 20. Given: PR 6 MG, MP 6 GR See margin.

P

R

Prove: Each diagonal of PRGM divides PRGM into two congruent triangles.

Proof 21. Given: PR 6 MG, MP 6 GR

Prove: PR > MG, MP > GR

(Hint: See Exercise 20.)

M

G

Since kPGM O kGPR (or kPMR O kGRM), then PR O MG

and MP O GR by CPCTC.

Test Prep

Multiple Choice Short Response

22. In the diagram, #RXW > # JXT. Which

statement is NOT necessarily true? C

A. & J > &R

B. &W > &T

C. WX > JX

D. RW > JT

23. Which is true by CPCTC? J

F. AC bisects BD H. &ABE > &EDC

G. &BAC > &DCA J. BC > DC

24. Which is not true by CPCTC? A

A. BE > DE

B. &BAC > &DAC

C. &BCA > &DCE D. AB > AD

R

T

X

W

J

B

A

E

C

D ABC ADC

Exercises 23?24

25. In the diagram, KB bisects &VKT and KV > KT.

K

a. What do you need to show in order to

conclude &KBV > &KBT ? State whether it is possible to show this and justify your answer.

V

B

T

b. Show that VB > TB. a?b. See margin.

Alternative Assessment

Have students work in pairs. Instruct each student to draw and label two congruent triangles, mark two of the corresponding parts congruent, and tell which parts must be proven congruent. Students then should exchange diagrams and take turns explaining how they can prove the triangles congruent.

Test Prep

Resources For additional practice with a variety of test item formats: ? Standardized Test Prep, p. 253 ? Test-Taking Strategies, p. 248 ? Test-Taking Strategies with

Transparencies

Mixed Review

Lesson 4-3

GO

for Help

What postulate or theorem can you use to prove the triangles congruent?

26. ASA

27.

AAS

Lesson 2-5 Lesson 2-3

28. The measure of an angle is 10 more than the measure of its supplement. Find the measures of both angles. 95; 85

If possible, use the Law of Detachment to draw a conclusion. If it is not possible to draw a conclusion, write not possible.

29. If two nonvertical lines are parallel, then their slopes are equal. Line m is nonvertical and parallel to line n.The slope of line m is the same as the slope of line n.

30. If a convex polygon is a quadrilateral, then the sum of its angle measures is 360. Convex polygon ABCDE has five sides. not possible

31. If a quadrilateral is a square, then it has four congruent sides. Quadrilateral ABCD has four congruent sides. not possible

lesson quiz, , Web Code: aua-0404

Lesson 4-4 Using Congruent Triangles: CPCTC 225

20. 1. PR n MG ; MP n GR (Given)

2. Draw PG . (2 pts. determine a line.)

3. lRPG O lPGM and lRGP O lGPM (If n lines, then alt. int. ' are O.)

4. kPGM O kGPR (ASA) A similar proof can be written if diagonal

RM is drawn.

25. [2] a. kKBV O kKBT; yes; SAS

b. CPCTC [1] one part correct

225

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