Abcd Parallelogram Circle Through And Intersect Cd Produce Necessary At Prove That Ae Ad

ABCD is a parallelogram.The circle through A,B and C intersect CD (produce if necessary) at E.Prove that AE = AD.

Answers:

Solution  :-

∠ABC + ∠AEC = 180⁰  (Opposite angles of cyclic quadrilateral)  .. . (i)
∠ADE + ∠ADC = 180⁰ (Linear pair)
But         ∠ADC = ∠ABC (Opposite angles of ||gm)
∴ ∠ADE + ∠ABC = 180⁰  .. (ii)
From equations (i) and (ii),we have
    ∠ABC + ∠AEC = ∠ADE + ∠ABC
 ⇒     ∠AEC = ∠ADE
 ⇒   AD = AE (sides opposite to equal angles)