Question

O is the center of the circle and AP is the tangent to the circle at point A. Ray AF is the bisector of angle BAC. Prove : seg AP congruent seg PE

Answer 1

To prove that segment AP is congruent to segment PE

• By the theorem the angles in the alternate segments are equal  

               angle PAC= angle ABC = y

•  AE bisects angle BAC

                      angle CAE = angle EAB = x

•      we know that exterior angle is equal to sum of the two interior angle

                                       angle PAE = x+y

                                      angle PEA = x+y

•     In  triangle APE

                 angle PAE = angle PEA

                       AP =PE