ABCD is a parallelogram. The circle through A, B and C intersects CD produced at E, prove that AE=AD

Answer : Given :ABCD is a parallelogram. The circle through A, B and C intersects CD produced at E

To prove : AE=AD

 

=> angle AED + angle ABC = 180  ... (1)  (Sum of opposite angles of cyclic quadrilaterals is 180)

=> angle ADE + angle ADC = 180  ... (2)  (linear pair)

=> angle ABC = angle ADC  ... ........(3)  (opposite angles of parallelogram are equal)

From (1) and (2)

=> angle AED + angle ABC = angle ADE + angle ADC

=> angle AED = angle ADE  (using (3))

In triangle AED,

 angle AED = angle ADE

=> AD = AE   (equal sides have equal angles opposite to them)

Hence Proved