Question & Answer

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that AP = CQ.

ANSWER Verified Verified
Hint: We will use congruence to make triangle APD and triangle BQC congruent and by c.p.c.t AP = CQ.

Complete step-by-step answer:

Now, in parallelogram, opposite sides are equal and parallel to each other. Also, the opposite angles are equal in a parallelogram.

So, in parallelogram ABCD, AD = BC. Also, AD is parallel to BC and diagonal BD acts as a transversal so, \[\angle {\text{ADP = }}\angle {\text{CBQ}}\]

Now, in triangle ADP and triangle CBQ,
AD = BC (Opposite sides)
\[\angle {\text{ADP = }}\angle {\text{CBQ}}\] (Alternate interior angles)
DP = BQ (Given)
So, $\Delta {\text{ADP }} \cong {\text{ }}\Delta {\text{CBQ}}$ By SAS property
So, both triangles are congruent to each other. Now, when two triangles are congruent to each other then corresponding parts are also equal to each other. So,
By Corresponding parts of Congruent triangle or C.P.C.T, we get
Hence proved.

Note: When we come up with such a type of solution, we have to apply the property of a given quadrilateral to solve the question. Also, to solve the problem, we should also use congruence of triangles. By using congruence, any question can be solved. It is the easiest method to find the solution of a given problem. Also, when you use congruence, always write the name of both the triangles in an exact manner, in which their congruent parts are equal to each other. Most students make this type of mistake when they write triangles in an improper manner, which leads them to a wrong answer.