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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