In the given figure, ABCD is a parallelogram. E is a point on DC. If AE bisects ∠A and BE bisects ∠B, then which of the following statements is correct?
AB = 2 BC
Let ∠A = 2x°
∠DAE = ∠EAB = x° …… (AE bisects ∠A)
ABCD is a parallelogram …… (given)
∠EAB = ∠AED = x° (alternate angles)
ADE is isosceles (base angles are equal to x°)
Therefore, DE = AD ………. (i)
∠B = 180° - 2x° ……(adjacent angles of a parallelogram)
∠ABE = ∠EBC = 90°- x° …… (AE bisects ∠A)
ABCD is a parallelogram …… (given)
∠EAB = ∠AED = 90°- x° (alternate angles)
BCEis isosceles (base angles are equal to x°)
Therefore, CE = CD ………. (ii)
But, AD = CD (opposite sides of a parallelogram are equal)
Therefore, AD = BC = DE = CE
But, DE + EC = DC = AB (opposite sides of a parallelogram)
AB = 2 BC
Hence (B)