Mathematics
If ABCD is a rectangle in which the diagonal BD bisects ∠B, then show that ABCD is a square.
Answer
Since, BD bisects ∠B.
∴ ∠1 = ∠2
Since, ∠B = 90°.
∠1 = ∠2 = 45°
∠4 = ∠1 = 45° (Alternate angles are equal)
∠3 = ∠2 = 45° (Alternate angles are equal)
In △ABD,
AB = AD (As sides opposite to equal angles are equal) ………(i)
In △CBD,
BC = CD (As sides opposite to equal angles are equal) ……….(ii)
and AD = BC, AB = CD (As opposite sides of rectangle are equal) …….(iii)
From (i), (ii) and (iii) we get,
AB = BC = CD = AD.
Since, all sides are equal and alternate sides are perpendicular to each other.
Hence, proved that ABCD is a square.
Related Questions
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P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.
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