1. Which side is included between R and F in FTR?

2. Which angles in STU include US?

Tell whether you can prove the triangles congruent by ASA or AAS. If you can, state a triangle congruence and the postulate or theorem you used. If not, write not possible.

3. 4. 5.

RF

S and U

Triangle Congruence by ASA and AASTriangle Congruence by ASA and AASGEOMETRY LESSON 4-3GEOMETRY LESSON 4-3

GHI PQRAAS

not possible ABX ACXAAS

4-3

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

4-4

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 because by definition, corresponding parts of congruent triangles are congruent.

The Basic Idea:The Basic Idea:

Given Information

Prove Triangles

CongruentShow

CorrespondingParts

Congruent

•SSS•SAS•ASA•AAS

CPCTC

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

4-4

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

Remember!

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

4-4

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

ExampleExample

A

B

C LJ

K

Is ABC JKL? YES

What’s the reason? SAS

Example continuedExample continued

A

B

C LJ

K

What other angles are congruent?B K and C L

What other side is congruent?

ABC JKL

BC KL

Example continuedExample continued

A

B

C LJ

K

What other angles are congruent?B K and C L

What other side is congruent?

ABC JKL

BC KL

Example

H J

KL

Prove: H KGiven: HJ || LK and JK || HL

Plan: Show JHL LKJ by ASA, then use CPCTC.

HJL KLJ (Alt Int s)

LJ LJ (Reflexive)

HLJ KJL (Alt Int s)

JHL LKJ (ASA)

H K (CPCTC)

QED

Example 2

A

R

T

M

S

Given: MS || TR and MS TR

Prove: A is the midpoint of MT.

SAM RAT (Vert. s)

SAM RAT (AAS)

Since MS || TR, M T (Alt. Int. s)

MS TR (Given)

MA AT (CPCTC)

A is the midpoint of MT (Def. midpoint)

Plan: Show the triangles are congruent using AAS, then MA =AT. By definition, A is the midpoint of segment MT.

Example 3

P

N L

M

Given: MP bisects LMN and LM NM

Prove: LP NP

NMP LMP (def. bis)

PMN PML (SAS)

QED

MP bis. LMN (Given)

LM NM (Given)

PM PM (Ref)

LP NP (CPCTC)

Given: AB DC, AD BC

Prove: A C

A B

CD

Statements Reasons

1. AB DC 1. Given

2. AD BC 2. Given

3. BD BD 3. Reflexive

4. ABD CDB 4. SSS

5. A C 5. CPCTC

A

B

CE

D

2. A D

3. ACB DCE

4. ACB DCE5. B E

1. AC DC (given)

(given)

(vert s)

(ASA)

(CPCTC)

Show B E

ProofsProofs

Ask: to show angles or segments congruent, what triangles must be congruent?

Then, how do you prove triangles congruent? (SSS, SAS, ASA, AAS)

Prove triangles congruent, then use CPCTC.

What other congruence statements can you prove from the

diagram, in which SL SR, and 1 2 are given? SC SC by the

Reflexive Property of Congruence, and LSC RSC by SAS.

3 4 by corresponding parts of congruent triangles are congruent.

When two triangles are congruent, you can form congruence statements about three pairs of corresponding angles and three pairs of corresponding sides. List the congruence statements.

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

4-4

Real-World Connection

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

(continued)

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

SL SR Given

SC SC Reflexive Property of Congruence

CL CR Other congruence statement

Sides:

1 2 Given 3 4 Corresponding Parts of

Congruent Triangles

CLS CRS Other congruence statement

Angles:

In the proof, three congruence statements are used, and one congruence statement is proven. That leaves two congruence statements remaining that also can be proved:CLS CRSCL CR

4-4

Quick Check

The Given states that DEG and DEF are right angles.

What conditions must hold for that to be true?

DEG and DEF are the angles the officer makes with the ground.

So the officer must stand perpendicular to the ground, and the ground must be level.

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

4-4

Quick Check

Real-World Connection

1. What does “CPCTC” stand for?

Use the diagram for Exercises 2 and 3.2. Tell how you would show ABM ACM.

3. Tell what other parts are congruent by CPCTC.

Use the diagram for Exercises 4 and 5.4. Tell how you would show RUQ TUS.

5. Tell what other parts are congruent by CPCTC.

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTCGEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

Corresponding parts of congruent triangles are congruent.

RQ TS, UQ US, R T

You are given two pairs of s, and AM AM by the Reflexive Prop., so ABM ACM by ASA.

4-4

You are given a pair of s and a pair of sides, and RUQ TUS because vertical angles are , so RUQ TUS by AAS.

AB AC, BM CM, B C

4-4

GEOMETRY LESSON 4-4GEOMETRY LESSON 4-4

In the diagram, JRC HVG.

1.List the congruent corresponding angles.

2.List the congruent corresponding sides.

You are given that TIC LOK.

3.List the congruent corresponding angles.

4.List the congruent corresponding sides.

(For help, go to Lesson 4-1.)

Using Congruent Triangles: CPCTCUsing Congruent Triangles: CPCTC

Check Skills You’ll Need

J H, R V, and C G

JR HV, RC VG, and JC HG

T L, I O, and C K

TI LO, IC OK, and CT KL