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 onetriangle are congruent to two angles andthe included side of another triangle, thenthe 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