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Question
In a quadrilateral ABCD, AB = AD and CB = CD.
Prove that:
- AC bisects angle BAD.
- AC is the perpendicular bisector of BD.
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Solution
Given: ABCD is quadrilateral,
AB = AD
CB = CD
To prove:
- AC bisects angle BAD.
- 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.
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