Similarity of Triangles | Moosa Academy

A triangle is the simplest polygon. Two triangles are similar when they have the same shape but not necessarily the same size. For triangles, a single test out of three is enough to prove similarity.

Theorem The three similarity criteria

Two triangles are similar if any one of these three conditions holds.

AA — two pairs of corresponding angles are equal.

SSS — the three pairs of corresponding sides are in the same ratio.

SAS — two pairs of sides are in the same ratio and the included angles are equal.

Case 1 AA — two equal angles

If two angles of one triangle equal two angles of another, the triangles are similar. The third angle matches automatically, since the angles of a triangle sum to 180°.

∠A = 60°, ∠B = 50° ⟹ ∠C = 70°

∠D = 60°, ∠E = 50° ⟹ ∠F = 70°

⟹ △ABC ~ △DEF (AA)

Case 2 SSS — equal side ratios

If the three pairs of corresponding sides are in the same ratio, the triangles are similar. That common ratio is the scale factor k.

A triangle with sides 150, 120, 90 and another with sides 75, 60, 45:

\frac{150}{75} = \frac{120}{60} = \frac{90}{45} = 2

⟹ all three ratios are equal, so the triangles are similar with $k = 2$ .

Case 3 SAS — two sides and the included angle

If two pairs of corresponding sides are in the same ratio and the angles between them are equal, the triangles are similar. The third side is not needed.

\dfrac{AB}{DE} = \dfrac{8}{16} = \dfrac{1}{2}

\dfrac{AC}{DF} = \dfrac{3}{6} = \dfrac{1}{2}

∠A = ∠D = 45° (the included angles)

⟹ the triangles are similar.

Summary

To prove two triangles similar, one of three tests is enough: AA, SSS, or SAS.
AA — two angles equal; the third is automatic since the angles sum to 180°.
SSS — the three side ratios are all equal to the scale factor k.
SAS — two sides in the same ratio with equal included angles.