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

(Given)

(Alternate interior angles)

(Opposite sides of a parallelogram)
Thus, by SideAngleSide 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 SideAngleSide 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
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,

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