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State whether the following statement is true or false:

In the following figure, if $AP$ is the bisector of $∠ A$ and $CQ$ is the bisector of $∠ C$ in parallelogram ABCD, then APCQ is a parallelogram.

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True

False

answer is A.

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Detailed Solution

Given, ABCD is a parallelogram where

AP

and CQ are angle bisectors of

∠ A

and

∠ C

respectively.
Question Image

AP and CQ are angle bisectors

⇒ ∠ B A P = 12 ∠ A

And,

∠ D C Q = 12 ∠ C

⇒

A = C

(Opposite angles of a parallelogram)
⇒

\frac{1}{2} A = \frac{1}{2} C

⇒

BAP = DCQ

Now,

∠ B A C = ∠ D C A

… (2) [

A B ∥ C D

, transversal AC, Alternate angles]
Subtracting (2) from (1),

⇒ ∠ B A P − ∠ B A C = ∠ D C Q − ∠ D C A

⇒ ∠ C A P = ∠ A C Q

As, these are alternate angles with AC as transversal,
⇒

A P ∥ Q C

Similarly,

P C ∥ A Q

Thus, APCQ is a parallelogram.
Hence the given statement is true. Hence, the correct option is 1.

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