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Diagonal AC of a parallelogram ABCD bisects $∠ A$ . then

(a) It bisects $∠ C$ also.

(b) ABCD is a rhombus.

State whether these statements are true or false.

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True

False

answer is A.

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Detailed Solution

Given,
ABCD = parallelogram
AC = diagonal AC bisects

∠ A

Thus,

∠ DAC = ∠ BAC

---- (1)
In the parallelogram we have,

∠ DAC = ∠ BCA

(interior angle)

∠ BAC = ∠ DCA

(Interior angle)
Therefore,

∠ DCA = ∠ BAC = ∠ DAC = ∠ BCA

Also, AC bisect

∠ C

. So,

∠ DAC = ∠ DCA

As, sides has equal length which are opposite to equal angle. Thus,

DA = DC

DA = BC and AB = CD

AB = BC = CD = DA

As, all sides are equal. Thus, ABCD is rhombus.

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