Key Points

Two geometric figures are called similar if they have the same shape but not necessarily the same size. All congruent figures are similar, but the converse is not always true.

Two polygons with the same number of sides are similar if (i) their corresponding angles are equal, and (ii) their corresponding sides are in the same ratio or proportion. For triangles, satisfying one condition implies the other.

If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then the other two sides are divided in the same ratio. In $\triangle ABC$, if $DE \parallel BC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. In $\triangle ABC$, if $\frac{AD}{DB} = \frac{AE}{EC}$, then $DE \parallel BC$.

If in two triangles, the corresponding angles are equal, then their corresponding sides are in the same ratio, and hence the two triangles are similar. This is known as the Angle-Angle-Angle (AAA) similarity criterion.

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This is a frequently used corollary of the AAA criterion.

If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal, and hence the two triangles are similar. This is the Side-Side-Side (SSS) criterion.

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the two triangles are similar. This is the Side-Angle-Side (SAS) criterion.

The similarity of triangles must be written with the correct correspondence of vertices. Writing $\triangle ABC \sim \triangle PQR$ implies that $\angle A = \angle P$, $\angle B = \angle Q$, $\angle C = \angle R$, and $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$.

If two triangles are similar, then the ratio of their corresponding sides is equal to the ratio of their corresponding medians and also to the ratio of their corresponding altitudes.