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Question
ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.
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Solution
Let us join AC.
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.
Similarly, it can be proved that BD bisects ∠B and ∠D as well.
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