Angle-Angle-Side (AAS): Definition, Examples, Chart, and Practice

What is Angle-Angle-Side?

Angle-Angle-Side is a triangle congruence rule.

It is often written as AAS.

AAS says that two triangles are congruent when two angles and one non-included side match in both triangles.

AAS congruence rule

Use AAS when the given side is not between the two given angles.

If two angles and a non-included side of one triangle match two angles and the matching non-included side of another triangle, the triangles are congruent.

Congruent means the triangles have the same size and the same shape.

What non-included side means

The included side is the side between the two given angles.

In AAS, the known side is not between the two known angles.

This is the main difference between AAS and ASA. ASA uses the side between the two angles. AAS uses a side away from that middle position.

AAS chart

This chart compares AAS with other triangle information. For the full rule list, start with triangle congruence first.

AAS works for triangle congruence. AAA only proves the triangles have the same shape, not always the same size.

How to use AAS

Step 1: Match the first pair of equal angles.

Step 2: Match the second pair of equal angles.

Step 3: Match the side that is not between those two angles.

Step 4: Write that the triangles are congruent by AAS.

Worked example

Problem: Triangle ABC and triangle DEF each have two matching angles. Angle A matches angle D, and angle C matches angle F. Side BC matches side EF. Can you prove the triangles are congruent?

Step 1: Two angle pairs match.

Step 2: The matching side is not the side between those two angles.

Step 3: That makes the rule Angle-Angle-Side.

Answer: The triangles are congruent by AAS.

Why AAS works

If two angles of a triangle are known, the third angle is forced because the three triangle angles add to 180°.

So AAS really gives enough information to lock the triangle shape and size.

The matching side fixes the scale, so the two triangles must be congruent.

AAS vs ASA

Both AAS and ASA use two angles and one side.

Use ASA when the side is between the two given angles.

Use AAS when the side is not between the two given angles.

Common mistakes

Do not use AAS if the side is between the two angles. That pattern is ASA.

Do not use AAA to prove congruence. Three matching angles can make similar triangles that are different sizes.

Do not match sides by guessing. Use tick marks, labels, or given statements to decide which side corresponds.

Quick practice

1. Two angles and a non-included side match in two triangles. The triangles are congruent by AAS.

2. Two angles and the included side match. That is ASA, not AAS.

3. Three angles match but no side is given. That is not enough to prove congruence.

4. If two angles are 48° and 72°, the third angle is 60° because triangle angles add to 180°.