In the figure, ABCD is a parallelogram and AP = CQ. Prove that PD = BQ. Prove also that the quadrilateral PBQD is a parallelogram.

Open in App

Solution

Given: A parallelogram ABCD

Also, P and Q are points AB and CD respectively such that AP = CQ.

In ΔAPD and ΔCQB,

AD = BC (Opposite sides of a parallelogram are equal)

∠A = ∠C (Opposite angles of a parallelogram are equal)

AP = CQ (Given)

As the two sides and the included angle are equal to the two sides and the included angle of the other triangle, ΔAPD ≅ ΔCQB

Corresponding parts of congruent triangles are congruent.

∴PD = BQ

Now, in quadrilateral PBQD:

AB = DC

⇒ AP + PB = CQ + QD

⇒ PB = QD (As AP = CQ)

Also, PD = BQ

We know that if opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

Thus, PBQD is a parallelogram.

Suggest Corrections

Similar questions

Q. In figure,

A B C D

is a parallelogram
Prove that

\frac{D P}{P Q} = \frac{D C}{B Q}

Q. P and Q are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CQ is parallel to AP. Prove also that AC bisects PQ.

In the adjoining figure, $A B C D$ is a parallelogram. If $P$ and $Q$ are points on $A D$ and $B C$ respectively such that $A P = \frac{1}{3} A D$ and $C Q = \frac{1}{3} B C$ , prove that $A Q C P$ is a parallelogram.

ABCD is a parallelogram. AP the bisector of ∠A and CQ the bisector of ∠C meet the opposite sides in P and Q, respectively. Prove that
AP || CQ.

P

and

Q

are point of trisection of the diagnoal

B D

of a parallelogram

A B C D

. Prove that

C Q | | A P

Join BYJU'S Learning Program