Learn

Congruent Triangles - ASA and AAS

Included Sides

So far, you have learned to prove triangles are congruent using the SSS and SAS postulates. You will learn two more methods now. Before we learn these methods, you need to learn a new concept. The concept is called included side. An included side is the side of a triangle that is between two angles. In other words, the included side has to be touching both angles. The included side of angles B and C is segment BC, because segment BC touches both angles B and C.

Triangle ABC. The sides of angle B are highlighted with arrows. The sides of angle C are highlighed with arrows. Segment BC is highlighted with an arrow.

Example #1

Let’s see if you can pick out the included side between the angles given. Name the included side between the given angles.

Triangle C E D.

1. ∠C and ∠D: Answer: segment CD

2. ∠E and ∠D: Answer: segment DE

3. ∠E and ∠C: Answer: segment CE

ASA Theorem

Great job, now you can move on to the new theorem. The ASA Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. The order of the letters ASA (angle, side, angle) helps you to remember that side has to be between the two angles.

Two triangles. Two angles and the included side of one triangle are congruent to two angles and an included side of other triangle.

AAS Theorem

Here is the last theorem you will need to learn about proving two triangles are congruent. The AAS Theorem states that if two angles and a nonincluded side of one triangle is congruent to two angles and a nonincluded side of another triangle, then the triangles are congruent. The order of the letters AAS (angle, angle, side) helps you to remember that side is not between the two angles.

Two triangles. Two angles and a non-included side of one triangle is congruent to two angles and a non-included side of the other triangle.

Example #2

Open Prove that two triangles are congruent using ASA And AAS Congruence Postulates in a new tab