On this page, we hope to clear up problems that you might
have with similar polygons. Similar polygons are
useful when you do stuff like enlarging a figure.
Scroll down or click one of the links below to start
understanding similar polygons!
Lines parallel to one side of a triangle
Quiz on Similar Polygons
Similar polygons are polygons for which all
corresponding angles are congruent and all corresponding sides
are proportional. Example:
1. Problem: Find the value of x, y, and the measure of angle P. Solution: To find the value of x and y, write proportions involving corresponding sides. Then use cross products to solve. 4 x 4 7 - = - - = - 6 9 6 y 6x = 36 4y = 42 x = 6 y = 10.5 To find angle P, note that angle P and angle S are corresponding angles. By definition of similar polygons, angle P = angle S = 86o.
The triangle, geometry's pet shape :-) , has a couple
of special rules dealing with similarity. They are outlined below.
1. Problem: Prove triangle ABE is similar to triangle CDE. Solution: Angle A and angle C are congruent (this information is given in the figure). Angle AEB and angle CED are congruent because vertical angles are congruent. Triangle ABE and triangle CDE are similar by Angle-Angle.
2. Side-Side-Side Similarity - If all pairs
of corresponding sides of two triangles are proportional, then
the triangles are similar.
2. Problem: Are the triangles shown in the figure similar? Solution: Find the ratios of the corresponding sides. UV 9 3 VW 15 3 -- = -- = - -- = -- = - KL 12 4 LM 20 4 The sides that include angle V and angle L are proportional. Angle V and angle L are congruent (the information is given in the figure). Triangle UVS and triangle KLM are similar by Side-Angle-Side.
What do parallel lines and triangles have to do with similar
polygons? Well, you can create similar triangles by
drawing a segment parallel to one side of a triangle in the triangle. This
is useful when you have to find the value of a triangle's side (or, in
a really scary case, only part of the
value of a side).
1. Problem: Find PT and PR
Take the Quiz on similar polygons. (Very useful to review or to see if you've really got this topic down.) Do it!