# PQ and RS are two equal and parallel line-segments. Any point M not lying on PQ or RS is joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove that line segments MN and PQ are equal and parallel to each other.

**Solution:**

Given, PQ and RS are two equal and parallel __line-segments__

Any point M not lying on PQ or RS is joined to Q and S.

The lines through P and R parallel to QM and SM meet at N.

We have to prove that the line segments MN and PQ are equal and __parallel__ to each other.

We know that the opposite sides of a __parallelogram__ are parallel and congruent.

PQ = RS and PQ|| RS --------- (1)

PQRS is a parallelogram.

We know that the sum of __interior angles__ lying on the same side of the transversal is always supplementary.

∠RPQ + ∠PQS = 180°

Now, ∠RPQ + ∠PQM + ∠MQS = 180° --------- (2)

Also, PN || QM

So, ∠NPQ + ∠PQM = 180°

Now, ∠NPR + ∠RPQ + ∠PQM = 180° ----------- (3)

Comparing (2) and (3),

∠MQS = ∠NPR ------------- (4)

Similarly, ∠MSQ = ∠NRP ---------- (5)

From (1), (4) and (5)

By ASA criteria, the triangles PNR and QMS are congruent.

By CPCTC,

NR = MS

PN = QM

So, PN || QM

Therefore, PQMN is a parallelogram

We know that the opposite sides of a parallelogram are parallel and congruent.

MN = PQ

NM || PQ

Therefore, it is proven that the line segments MN and PQ are equal and parallel to each other.

**✦ Try This:** Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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

**NCERT Exemplar Class 9 Maths Exercise 8.4 Sample Problem 1**

## PQ and RS are two equal and parallel line-segments. Any point M not lying on PQ or RS is joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove that line segments MN and PQ are equal and parallel to each other.

**Summary:**

PQ and RS are two equal and parallel line-segments. Any point M not lying on PQ or RS is joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. It is proven that line segments MN and PQ are equal and parallel to each other

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