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

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

1. APCQ is a parallelogram

Solution:

1. In ,

• (Given)

• (Alternate interior angles)

• (Opposite sides of a parallelogram)

Thus, by Side-Angle-Side congruence condition.

2. By Corresponding parts of congruent triangles as .

3. In ,

• (Given)

• (Alternate interior angles)

• AB = CD (Opposite sides of a parallelogram)

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

4. By Corresponding Parts of Congruent Triangles as .

5. 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 .

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