Parallel Lines        Congruent Tri.        Congruent R. Tri.        Isosc. and Equil.        Quadrilaterals        Parallelograms        Ratios    Similar Polygons        Special Triangles       Circles        Area        Coordinate Geo.        Triangle Ineq.        Solids        Computer Fun

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! Special similarity rules for triangles 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: Many times you will be asked to find the measures of angles and sides of figures.  Similar polygons can help you out. 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.  Angle-Angle Similarity - If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.  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. 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. 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. Back to Top  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). 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. 1.   Problem: Find PT and PR 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 Back to Top  Take the Quiz on similar polygons.  (Very useful to review or to see if you've really got this topic down.)  Do it!