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Holt Geometry

4-6 Triangle Congruence: CPCTC4-6 Triangle Congruence: CPCTC

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

4-6 Triangle Congruence: CPCTC

Warm Up

1. If ∆ABC ∆DEF, then A ? and BC ? .

2. List methods used to prove two triangles congruent.

D EF

SSS, SAS, ASA, AAS, HL

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SWBAT use CPCTC to prove parts of triangles are congruent.

Objective

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CPCTC

Vocabulary

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CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent.

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SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.

Remember!

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Example 1: Engineering Application

A and B are on the edges of a ravine. What is AB? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal.

Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so AB = 18 mi.

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Check It Out! Example 1

A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles.

Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so JK = 41 ft.

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Example 2: Proving Corresponding Parts Congruent

Prove: XYW ZYW

Given: YW bisects XZ, XY YZ.

Z

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Example 2 Continued

WY

ZW

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Prove: PQ PS

Given: PR bisects QPS and QRS.

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Check It Out! Example 2 Continued

PR bisects QPS

and QRS

QRP SRP

QPR SPR

Given Def. of bisector

RP PR

Reflex. Prop. of

∆PQR ∆PSR

PQ PS

ASA

CPCTC

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Work backward when planning a proof. To show that ED || GF, look for a pair of angles that are congruent.

Then look for triangles that contain these angles.

Helpful Hint

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Example 3: Using CPCTC in a Proof

Prove: MN || OP

Given: NO || MP, N P

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5. CPCTC5. NMO POM

6. If Alt. Int. s then // lines

4. AAS (1,3,2)4. ∆MNO ∆OPM

3. Reflex. Prop. of

2. If // lines, then Alt. Int. s 2. NOM PMO

1. Given

ReasonsStatements

3. MO MO

6. MN || OP

1. N P; NO || MP

Example 3 Continued

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Prove: KL || MN

Given: J is the midpoint of KM and NL.

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Check It Out! Example 3 Continued

5. CPCTC5. LKJ NMJ

6. If Alt. Int. s then // lines

4. SAS (2, 3, 2)4. ∆KJL ∆MJN

3. Vert. s 3. KJL MJN

2. Def. of mdpt.

1. Given

ReasonsStatements

6. KL || MN

1. J is the midpoint of KM and NL.

2. KJ MJ, NJ LJ

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Lesson Quiz: Part I

1. Given: Isosceles ∆PQR, base QR, PA PB

Prove: AR BQ

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4. Reflex. Prop. of 4. P P

5. SAS (2, 4, 3)5. ∆QPB ∆RPA

6. CPCTC6. AR = BQ

3. Given3. PA = PB

2. Def. of Isosc. ∆2. PQ = PR

1. Isosc. ∆PQR, base QR

Statements

1. Given

Reasons

Lesson Quiz: Part I Continued

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Lesson Quiz: Part II

2. Given: X is the midpoint of AC . 1 2

Prove: X is the midpoint of BD.

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Lesson Quiz: Part II Continued

6. CPCTC

5. ASA (1, 3, 4)5. ∆AXD ∆CXB

7. Def. of mdpt.7. X is mdpt. of BD.

4. Vert. s 4. AXD CXB

3. Def of midpoint3. AX CX

2. Given2. X is mdpt. of AC.

1. Given1. 1 2

ReasonsStatements

6. DX BX

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