Similar Polygons | Moosa Academy

A polygon is any closed figure with three or more straight sides. Two polygons are similar when they have the same shape but not necessarily the same size.

Theorem Similarity of Polygons

Two polygons are similar when both conditions hold: their corresponding angles are equal, and the ratios of their corresponding sides are all equal.

Corresponding angles are equal, and every pair of corresponding sides shares one ratio:

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \cdots = k

One condition alone is not enough for a general polygon. A square and a rectangle both have four 90° angles, yet their side ratios differ — so they are not similar.

Example Two squares

Square 1 has side 4 and square 2 has side 8. Are they similar?

Side ratio

=\dfrac{8}{4}=2

, the same for every side.

All angles of both squares are 90°.

⟹ the squares are similar,

k=2

Theorem Special case: triangles

For triangles, one condition is enough. If two pairs of corresponding angles are equal, the triangles are automatically similar.

The angles of a triangle sum to 180°.

If two angles match, the third must match too.

Equal angles then force equal side ratios.

⟹ equal angles alone prove similarity.

Triangles ABC and DEF share all three angles (90°, 60°, 30°) — matching arc counts mark the corresponding pairs.

Summary

Similarity keeps the shape and changes only the size.
General polygons need both conditions — equal angles and equal side ratios.
Triangles are a special case: one condition is enough, and the other follows automatically.
All squares are similar to one another; rectangles are not always similar.