4-3 Congruent Triangles

You identified and used congruent angles.

• Name and use corresponding parts of congruent polygons.

• Prove triangles congruent using the definition of congruence.

• If two geometric figures have exactly the same shape and size, they are congruent.

• Un two congruent polygons, all of the parts of one polygon are congruent to the corresponding parts or matching parts of the other polygon. These corresponding parts include corresponding angles and corresponding sides.

Vocabulary

The Right MatchCongruent Not congruent

A

B

C

D

EF

Corresponding sides and angles

Corresponding Angles

RA Corresponding sides

A B

C

R S

T

SB TC AB RS

BC ST

AC RT

A

B

C

H

JK

Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.

Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE RTPSQ.

Sides:

Angles:

The support beams on the fence form congruent triangles. In the figure ΔABC ΔDEF, which of the following congruence statements correctly identifies corresponding angles or sides?

A.

B.

C.

D.

The phrase “if and only if” in the congruent polygon definition means that both the conditional and its converse are true. So, if two polygons are congruent, then their corresponding parts are congruent. For triangles, Corresponding parts of congruent triangles are congruent, or CPCTC.

C P CTC

O P CPCTC

mO = mP Definition of congruence

6y – 14 = 40 Substitution

In the diagram, ΔITP ΔNGO. Find the values of x and y.

6y = 54 Add 14 to each side.

y = 9 Divide each side by 6.

NG = IT Definition of congruence

x – 2y = 7.5 Substitution y = 9

x – 18 = 7.5 x = 25.5, y = 9

CPCTC

A. x = 4.5, y = 2.75

B. x = 2.75, y = 4.5

C. x = 1.8, y = 19

D. x = 4.5, y = 5.5

In the diagram, ΔFHJ ΔHFG. Find the values of x and y.

ARCHITECTURE A drawing of a tower’s roof is composed of congruent triangles all converging at a point at the top. If IJK IKJ and mIJK = 72, find mJIH.

mIJK + mIJK + mJIK = 180 Substitution

72 + 72 + mJIK = 180 Substitution

144 + mJIK = 180 Simplify.

mJIK = 36 Subtract 144 from each side.

mJIH = 36 Third Angles Theorem

Answer: mJIH = 36

mIJK + mIKJ + mJIK = 180 Triangle Angle-Sum Theorem

ΔJIK ΔJIH Congruent Triangles

Write a two-column proof.

Prove: ΔLMN ΔPON

2. LNM PNO 2. Vertical Angles Theorem

Statements Reasons

3. M O

3. Third Angles Theorem

4. ΔLMN ΔPON

4. CPCTC

1. Given1.

Find the missing information in the following proof.

Prove: ΔQNP ΔOPN

ReasonsStatements

3. Q O, NPQ PNO 3. Given

5. Definition of Congruent Polygons5. ΔQNP ΔOPN

4. _________________4. QNP ONP ?

2. 2. Reflexive Property ofCongruence

1. 1. Given

Third angle theorem

Like congruence of segments and angles, congruence of triangles is reflexive, symmetric, and transitive.

4-3 AssignmentPage 259, 10, 12, 13-16, 18-20