# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 5

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Q-9 In parallelogram ABCD, two points P and Q are taken on diagonal BD such that (see Fig.). Show that:

APCQ is a parallelogram

Solution:

In,

(Given)

(Alternate interior angles)

(Opposite sides of a parallelogram)

Thus, by Side-Angle-Side congruence condition.

By Corresponding parts of congruent triangles as .

In ,

(Given)

(Alternate interior angles)

AB = CD (Opposite sides of a parallelogram)

Thus, ΔAQB ≅ ΔCPD by Side-Angle-Side congruence condition.

By Corresponding Parts of Congruent Triangles as.

The diagonal of a parallelogram bisect each other.

Given

....equation (1)

Also,………….equation (2) (diagonal of a parallelogram bisect each other)

From equation (1) and (2), APCQ is parallelogram

Q-10 ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on on diagonal BD (see Fig.).Show that

Solution:

Given,

ABCD is parallelogram

AP and CQ are perpendiculars from A and C on diagonal BD

Solution (i)

In and ,

(Opposite side of parallelogram ABCD)

(Alternate interior angles)

Now,

(Equal to right angles as AP and CQ are perpendiculars)

(ABCD is a parallelogram)

Thus, by Angle-Angle-Side congruence condition.

Solution (ii)

by Corresponding Parts of Congruent Triangles as .