Similar figures have equal corresponding angles and one common ratio between all corresponding lengths. A scale factor k from the first figure to the second means “second length divided by first length”. Reduction has 0<k<1; enlargement has k>1; k=1 gives congruence. **Triangle test** | **Required information** AA | Two equal pairs of corresponding angles. SSS similarity | All three corresponding side ratios equal. SAS similarity | Two corresponding side ratios equal, with equal included angles. Example In ABC PQR, AB=6 cm and PQ=15 cm. The scale factor from ABC to PQR is 156=2.5. If AC=8 cm, then PR=20 cm. The reverse scale factor is 0.4. All corresponding lengths, including heights, diagonals and perimeters, scale by k. For general polygons, equal angles alone or proportional sides alone need not establish similarity.

Similar figures have equal corresponding angles and one common ratio between all corresponding lengths. A scale factor \(k\) from the first figure to the second means “second length divided by first length”. Reduction has \(0<k<1\); enlargement has \(k>1\); \(k=1\) gives congruence.

In \(\triangle ABC\sim\triangle PQR\), \(AB=6\text{ cm}\) and \(PQ=15\text{ cm}\). The scale factor from \(ABC\) to \(PQR\) is \(\dfrac{15}{6}=2.5\). If \(AC=8\text{ cm}\), then \(PR=20\text{ cm}\). The reverse scale factor is \(0.4\).

All corresponding lengths, including heights, diagonals and perimeters, scale by \(k\). For general polygons, equal angles alone or proportional sides alone need not establish similarity.

Similarity and corresponding lengths

Similarity and corresponding lengths

Triangle test	Required information
AA	Two equal pairs of corresponding angles.
SSS similarity	All three corresponding side ratios equal.
SAS similarity	Two corresponding side ratios equal, with equal included angles.