Geometry: Similar Polygons

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. Read on or follow any of the links below to start understanding similar polygons!

Special similarity rules for triangles
Lines parallel to one side of a triangle
Quiz on similar polygons

What are Similar Polygons?

Similar polygons are polygons for which all corresponding angles are congruent and all corresponding sides are proportional. Example: Example Figure

Many times you will be asked to find the measures of angles and sides of figures. Similar polygons can help you out.

Example:

1.  Problem: Find the value of x, y, and the measure of angle P in the
             accompanying figure.
   Solution: To find the value of x and y, write proportions involving
             corresponding sides.  The 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 = 86°.

Special Similarity Rules for Triangles

The triangle, geometry's pet shape, :-) has a couple of special rules dealing with similarity. They are outlined below.

1. Angle-Angle Similarity - If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Example:

1.  Problem: Prove triagnle ABE is similar to triangle CDE in the
             accompanying figure.
   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.

3. Side-Angle-Side Similarity - If one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional, then the two triangles are similar.

Example:

1.  Problem: Are the triangles shown in the accompanying 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.

Parallel Lines and Triangles

What to 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).

The theorem that lets us do that says if a segment is parallel to one side of a triangle and intersects the other sides in two points, then the triangle formed is similar to the original triangle. Also, when you put a parallel line in a triangle, as the theorem above describes, the sides are divided proportionally.

Example:

1.  Problem: Find PT and PR in the accompanying figure.
   Solution: 4    x
             - = --     Because the sides are divided
             7   12     proportionally when you draw a
                        parallel line to another side.

             7x = 48    Cross products.

             x = 48/7

             PT = 48/7
             PR = 12 + 48/7 = 132/7

Take the quiz on similar polygons. The quiz is very useful for either review or to see if you've really got the topic down.

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