AAA Similarity Rule

The AAA Similarity Rule states that if all three angles of one triangle are equal to the corresponding three angles of another triangle, then the triangles are similar.

They will have the same shape but may differ in size.

 Triangle 1: ∠A, ∠B, ∠C
 Triangle 2: ∠D, ∠E, ∠F
If ∠A=∠D, ∠B=∠E, ∠C=∠F → Triangles are similar

AAA Rule:

If in two triangles, the corresponding angles are equal, then the two triangles are similar.

Symbolically, if:

• \(\angle A = \angle D\)
• \(\angle B = \angle E\)
• \(\angle C = \angle F\)

then:

\(\triangle ABC \sim \triangle DEF\)

In a triangle:

• The sum of angles is always \(180^\circ\).
• If two angles are equal, the third is automatically equal.

This forces the two triangles to have the same shape. The side lengths may differ, but the ratio of sides remains the same.

       A                  D
      / \                / \
     /   \              /   \
    B-----C            E-----F
∠A = ∠D, ∠B = ∠E, ∠C = ∠F → Similar

The AAA rule does not prove congruence because the triangles may be of different sizes. It only proves similarity.

Two triangles can have the same angles but different side lengths, so they cannot be congruent unless their sides also match.

AAA similarity is used in many real-life applications, such as:

• Scaling geometric drawings and models,
• Shadow and height calculations,
• Maps and blueprint designs,
• Optics (light ray triangles remain similar when reflected).