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AAA similarity test

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

If two triangles have two pairs of equal angles, then their third pair is also equal because the angle sum of a triangle is 180°. Thus two such triangles are called equi-angular, and the test is often referred to as the AAA similarity test.

Two equi-angular triangle
Detailed description

Proof

Let \(\triangle ABC\) and \(\triangle PQR\) be triangles with \(\angle A = \angle P = \alpha\) and \(\angle B = \angle Q = \beta\).

One triangle.
Detailed description

Enlarge \(\triangle ABC\) to \(\triangle A^\prime B^\prime C^\prime\) by a suitable enlargement factor so that \(A^\prime B^\prime = PQ\).

Then \(\angle A^\prime = \alpha\) and \(\angle B^\prime = \beta\) because angles are preserved by enlargements.

Hence \(\triangle A^\prime B^\prime C^\prime\) is congruent to \(\triangle PQR\) by the AAS congruence test. So \(\triangle ABC\) is similar to \(\triangle PQR\), because an enlargement of \(\triangle ABC\) is congruent to \(\triangle PQR\).

 

Example 1

Prove that \(\triangle ABC\) is similar to \(\triangle DEC\).

Two triangles
Detailed description

Solution

In the triangles \(ABC\) and \(DEC\)

\(AB\ \text{is parallel to}\ ED\\ \angle CAB = \angle CDE\ (\text{corresponding angles})\\ \angle CBA = \angle CED\ (\text{corresponding angles})\\ \angle ACB = \angle DCE\ (\text{vertically opposite})\)

So \(\triangle ABC\) is similar to \(\triangle DEC\) (AAA similarity test).

Note that it is only necessary to show that two pairs of matching angles are equal. The third pair are necessarily equal because the angles of a triangle add to 180°.