Question

In a quadrilateral ABCD, AB = AD and CB = CD.

Prove that:

1. AC bisects angle BAD.
2. AC is the perpendicular bisector of BD.

Answer 1

Given: ABCD is quadrilateral,

AB = AD

CB = CD

To prove:

1. AC bisects angle BAD.
2. AC is the perpendicular bisector of BD.

Proof:

In ΔABC and ΔADC,

AB = AD                    ...(given)

CB = CD                    ...(given)

AC = AC                    ...(Common side)

ΔABC ≅ ΔADC          ...(SSS)

∠BAD = ∠DAO         ...(AC bisects A)

Therefore, AC bisects ∠BAD

OD = OB

OA = OC                 ...(diagonals bisect each other at O)

Thus, AC is perpendicular bisector of BD.

Hence, proved.