Q. 74.0( 404 Votes )

# ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.

Answer :

**Let us join AC.**

To Prove: ∠1 = ∠4, ∠2 = ∠3

Proof:

In ΔABC,

BC = AB (Sides of a rhombus are equal to each other)

∠1 = ∠2 (Angles opposite to equal sides of a triangle are equal)

However,

∠1 = ∠3 (Alternate interior angles for parallel lines AB and CD)

∠2 = ∠3

Therefore, AC bisects ∠C

Also,

∠2 = ∠4 (Alternate interior angles for || lines BC and DA)

∠1 = ∠4

Therefore,

AC bisects ∠A

Hence, Proved.**Similarly, it can be proved that BD bisects B and D as well.**

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