Question

In the given figure, ABCD is a parallelogram. E is a point on DC. If AE bisects $\angle$A and BE bisects $\angle$B, then BC : AB = ___.

• A. 2 : 1
• B. 1 : 2
• C. 1 : 3
• D. 3 : 1

Answer 1

The correct option is B

1 : 2

Let $\angle$A = 2x°

$\angle$DAE = $\angle$EAB = x° …… (AE bisects $\angle$A)

ABCD is a parallelogram …… (given)

$\angle$EAB = $\angle$AED = x° (alternate angles)

ADE is isosceles (base angles are equal to x°)

Therefore, DE = AD ………. (i)

$\angle$B = 180° - 2x° ……(adjacent angles of a parallelogram)

$\angle$ABE = $\angle$EBC = 90°- x° …… (AE bisects $\angle$A)

ABCD is a parallelogram …… (given)

$\angle$EAB = $\angle$AED = 90°- x° (alternate angles)

BCEis isosceles (base angles are equal to x°)

Therefore, CE = CB ………. (ii)

But, AD = CB (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

BC : AB = 1 : 2