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

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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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☛ Related Questions:

• Prove that a diagonal of a parallelogram divides it into two congruent triangles
• Show that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, . . . .
• A diagonal of a parallelogram bisects one of its angle. Prove that it will bisect its opposite angle . . . .