1

Objectives

• Define congruent polygons

• Prove that two triangles are congruent using SSS, SAS, ASA, and AAS shortcuts

2

Definition of Congruence

• Congruent figures have the same shape and size

• Congruent polygons have congruent corresponding parts – matching sides and angles

3

Proving Two Triangles Congruent

• Using definition: all corresponding sides and angles congruent

• Using shortcuts: SSS, SAS, ASA, AAS

4

Side-Side-Side (SSS) Postulate

If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

ΔGHF ≅ ΔPQR

5

Side-Angle-Side (SAS) Postulate

If two sides and the included angle of onetriangle are congruent to two sides and theincluded angle of another triangle, then thetwo triangles are congruent. ΔBCA ≅ ΔFDE

6

Angle-Side-Angle (ASA) Postulate

If two angles and the included side of one

triangle are congruent to two angles and

the included side of another triangle, then

the two triangles are congruent.

ΔHGB ≅ ΔNKP

7

Angle-Angle-Side (AAS) Theorem

If two angles and a nonincluded side of onetriangle are congruent to two angles and thecorresponding nonincluded side of anothertriangle, then the triangles are congruent. ΔCDM ≅ ΔXGT

8

SSS Example

• AB CD and BC DA (Given)

• AC AC (Reflexive Property of Congruence)

• ∆ABC ∆CDA by SSS.

Explain why ∆ABC ∆CDA.

9

SAS Example

• XZ VZ and WZ YZ (Given)

• ∠XZY ∠VZW (Vertical angles are )

• ∆XZY ∆VZW (SAS)

Explain why ∆XZY ∆VZW

10

ASA Example

Explain why

∆KLN ∆MNL

KL MN (Given)

KL || MN (Given)

∠KLN ∠MNL (Alternate Interior Angles are )

LN LN (Reflexive Property)

∆KLN ∆MNL (SAS)

11

AAS Example

JL bisects ∠KJM (Given)

∠KJL ∠MJL (Defn. of angle bisector)

∠K ∠M (Given)

JL JL (Reflexive Property)

∆JKL ∆JML (AAS)

Given: JL bisects ∠KJM. Explain why ∆JKL ∆JML