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Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional.

The three similarity shortcuts

AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another, the triangles are similar.

SSS (Side-Side-Side): If all three pairs of corresponding sides are in the same ratio, the triangles are similar.

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$

SAS (Side-Angle-Side): If two sides are proportional and the angle between them is congruent, the triangles are similar.

Why it matters

Similar triangles let us measure things we can't reach — the height of a flagpole from its shadow, or the width of a river from a few measurements on one side.

Example 1

In $\triangle ABC$ and $\triangle DEF$, $\angle A \cong \angle D$ and $\angle B \cong \angle E$. Are the triangles similar?

Yes — by AA. Two pairs of congruent angles are enough; the third pair must match too.

Example 2

$\triangle ABC$ has sides $6, 8, 10$. $\triangle DEF$ has sides $9, 12, 15$. Are they similar?

Check the ratios: $\frac{6}{9} = \frac{8}{12} = \frac{10}{15} = \frac{2}{3}$

All ratios equal $\frac{2}{3}$, so the triangles are similar by SSS.

Example 3 (try the next step yourself)

A flagpole casts a $24$ ft shadow while a $5$ ft tall student casts a $3$ ft shadow. Set up the proportion — then solve for the flagpole's height.