farid zen

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.
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