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 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°.
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 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.
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.Back to top.
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 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 = 132/7Back to top.
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.