Chapter 4Congruent Triangles

Congruent figures- ◦ They have exactly the same shape.◦ All parts of one figure are congruent to the

corresponding parts of the other figure.◦ Corresponding sides and angles are congruent.

4.2 Apply Congruence and Triangles

A

B

C D F

E

Name the congruent triangles.

Example

A

B

C

D

F

E

Find x and y.

Example

ABCD FGHK

4x +2y 44

136

80

3x -6 in.9 in.

AD

CB G H

KF

If 2 angles of one triangle are congruent to 2 angles of another triangle, then the third angles are also congruent.

Third Angle Theorem

ExampleFind mYXW.

40

35X

ZW

Y

ExampleProve: ACD CAB

Given: AD CB, DC BA, ACD CAB, CAD ACBA

C

B

D

Side-Side-Side Congruence Postulate- If 3 sides of one triangle are congruent to 3 sides of a second triangle, then the two triangles are congruent.

4.3 Prove Triangles Congruent by SSS

K

L

M

N

Side- Angle- Side Congruence Postulate – If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of a second triangle, then the 2 triangles are congruent.

4.4 Prove Triangles Congruent by SAS and HL

Given: KMMN,ML bisects KMNProve: KML NML

L N

M

K

R is the center of the circle. Based on the diagram, what can you conclude about ∆URT and ∆SRT ?

Example

R

S T

U

Why SSA does not work.

CD

B

A

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the 2 triangles are congruent.

Hypotenuse Leg Congruence Theorem

A

B C

D

F E

Proof Prove: WYZ XZY

Given: WY XZ, WZ ZY, XY ZY

W

Z Y

X

Angle- Side- Angle Congruence Postulate- If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a second triangle, then the 2 triangles are congruent.

4.5 Prove Triangles Congruent by ASA and AAS

Angle- Angle- Side Congruence Theorem- If 2 angle and a non-included side of one triangle are congruent to 2 angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

Right Angle Congruence Theorem- All right angles are congruent.

Given: CBFCDF, BFFDProve: ABF EDF

F

C

A E

B D

CPCTC- Corresponding Parts of Congruent Triangles are Congruent

4.6 Use Congruent Triangles