# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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:

1. APCQ is a parallelogram

Solution:

1. In ,
2. (Given)
3. (Alternate interior angles)
4. (Opposite sides of a parallelogram)

Thus, by Side-Angle-Side congruence condition.

1. By Corresponding parts of congruent triangles as .
2. In ,
3. (Given)
4. (Alternate interior angles)
5. AB = CD (Opposite sides of a parallelogram)

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

1. By Corresponding Parts of Congruent Triangles as .
2. The diagonal of a parallelogram bisect each other.
3. Given
4. … 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 .