Similar Polygons

Two polygons with the same shape are called similar polygons. The symbol for “is similar to” is ∼. Notice that it is a portion of the “is congruent to” symbol, ≅. When two polygons are similar, these two facts both must be true:

  • Corresponding angles are equal.
  • The ratios of pairs of corresponding sides must all be equal.

 

In Figure 1, quadrilateral ABCD ∼ quadrilateral EFGH.

Figure 1 Similar quadrilaterals.


This means: m ∠ A = m ∠ Em ∠ B = m ∠ Fm ∠ C = m ∠ Gm ∠ D = m ∠ H, and 

It is possible for a polygon to have one of the above facts true without having the other fact true. The following two examples show how that is possible.

In Figure 2, quadrilateral QRST is not similar to quadrilateral WXYZ.



Figure 2 Quadrilaterals that are not similar to one another.



Even though the ratios of corresponding sides are equal, corresponding angles are not equal (90° ≠ 120°, 90° ≠ 60°).

In Figure 3, quadrilateral FGHI is not similar to quadrilateral JKLM.



Figure 3 Quadrilaterals that are not similar to one another.



Even though corresponding angles are equal, the ratios of each pair of corresponding sides are not equal (3/3≠5/3).

Example 1: In Figure 4, quadrilateral ABCD ∼ quadrilateral EFGH. (a) Find m ∠ E. (b) Find x.



Figure 4 Similar quadrilaterals.



(a) m ∠ E = 90° (∠ E and ∠ A are corresponding angles of similar polygons, and corresponding angles of similar polygons are equal.)

(b) 9/6 = 12/ x (If two polygons are similar, then the ratios of each pair of corresponding sides are equal.)