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4-7 Triangle Congruence: CPCTC

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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4-7 Triangle Congruence: CPCTC Remember!

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

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4-7 Triangle Congruence: CPCTC 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.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC 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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4-7 Triangle Congruence: CPCTC Example 2: Proving Corresponding Parts Congruent

Given: YW bisects XZ, XY  YZ.

Prove:



XYW

 

ZYW

Z

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4-7 Triangle Congruence: CPCTC Example 2 Continued

ZW WY

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC Check It Out!

Example 2

Given: PR bisects 

QPS and



QRS

.

Prove: PQ 

PS

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4-7 Triangle Congruence: CPCTC Check It Out!

Example 2 Continued

PR bisects 

QPS

and 

QRS

Given

RP



PR

Reflex. Prop. of  ∆PQR  ∆PSR ASA

PQ



PS

CPCTC

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

QRP

 

SRP



QPR

 

SPR

Def. of  bisector

4-7 Triangle Congruence: CPCTC Check It Out!

Example 3

Given: J is the midpoint of KM and NL. Prove: KL || MN

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4-7 Triangle Congruence: CPCTC Example 4: Using CPCTC In the Coordinate Plane

Given: D(–5, –5), E(–3, –1), F(–2, –3), G( – 2, 1), H(0, 5), and I(1, 3)

Prove:



DEF

 

GHI

Step 1 Plot the points on a coordinate plane.

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4-7 Triangle Congruence: CPCTC

Step 2 Use the Distance Formula to find the lengths of the sides of each triangle.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

So DE  GH, EF  HI, and DF  GI. Therefore ∆DEF  by CPCTC.

∆GHI by SSS, and 

DEF

 

GHI

Holt McDougal Geometry