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# Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP = CQ (Fig. 8.6). Show that AC and PQ bisect each other.

**Solution:**

Given, ABCD is a __parallelogram__

The points P and Q lie on the opposite sides AB and CD of the parallelogram

Given, AP = CQ

We have to show that AC and PQ bisect each other.

Considering triangles AMP and CMQ,

We know that __alternate interior angles__ are equal.

So, ∠PAM = ∠QCM

Given, AP = CQ

Also, the alternate interior angles ∠MPA and ∠MQC are equal.

So, ∠MPA = ∠MQC

__SAS__ criterion states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are __congruent__, then the triangles are congruent.

By SAS criteria, the triangles ANP and CMQ are congruent.

The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are congruent, then their __corresponding sides__ and angles are also congruent or equal in measurements.

By CPCTC,

AM = CM

PM = MQ

This implies AC and BQ bisect each other at M.

Therefore, AC and PQ bisect each other.

**✦ Try This: **In parallelogram ABCD, two point P and Q are taken on diagonal BD such that DP=BQ. Show that APCQ is a parallelogram.

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.3 Problem 9**

## Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP = CQ (Fig. 8.6). Show that AC and PQ bisect each other.

**Summary:**

Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP = CQ (Fig. 8.6). It is shown that AC and PQ bisect each other

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