Similar Triangles

Two equal angles force the same shape — then side ratios do the rest.

The idea

Two figures are similar when they have the same shape, though not necessarily the same size — one is a scaled copy of the other.

Definition (Similar triangles).

Two triangles are similar when every pair of corresponding angles is equal and every pair of corresponding sides is in one common ratio, the scale factor.

Multiplying any side of the first triangle by the scale factor gives the matching side of the second. For triangles the test for similarity is short.

Theorem (AA criterion).

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

The notation records the correspondence. $\triangle ABC \sim \triangle DEF$ asserts that $A$ matches $D$, $B$ matches $E$, and $C$ matches $F$, and corresponding sides are read off in the same order: $AB$ with $DE$, $BC$ with $EF$, $AC$ with $DF$.

Ways to work on it

• Walkthrough. The AA criterion, the scale factor, and a first proportion.
• Practice. Solve for an unknown side from an AA similarity.
• Hardest. Indirect measurement: find a flagpole's height from shadows.

Not sure where to start? Take the ten-question placement test.