NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 5
Q9 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,
Thus, by SideAngleSide congruence condition.

By Corresponding parts of congruent triangles as .

In ,
Thus, ΔAQB ≅ ΔCPD by SideAngleSide congruence condition.

By Corresponding Parts of Congruent Triangles as.

The diagonal of a parallelogram bisect each other.
Also,………….equation (2) (diagonal of a parallelogram bisect each other)
From equation (1) and (2), APCQ is parallelogram
Q10 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,
Solution (i)
In and ,
Now,

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

(ABCD is a parallelogram)

Thus, by AngleAngleSide congruence condition.
Solution (ii)
by Corresponding Parts of Congruent Triangles as .