NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 6
A parallelogram is a quadrilateral with opposite sides parallel
Trapezium is a quadrilateral with one pair of sides parallel
Q11 In and , , . Vertices A, B, C are joined to vertices D, E, F respectively (see figure).show that

Quadrilateral ABCD is a parallelogram

Quadrilateral BEFC is a parallelogram

and

Quadrilateral ACFD is a parallelogram


Solution:
Given,

and




.

and (Given)
So, quadrilateral ABED is a parallelogram because: one of the two pairs of opposite sides of a quadrilateral are both equal and parallel to each other.

Again and .
Therefore, quadrilateral BEFC is a parallelogram.

ABED is a parallelogram therefore,

And ……equation (1) (Opposite sides of a parallelogram are equal)

Therefore, BEFC is a parallelogram.

Also, and ……..equation (2) (Opposite sides of a parallelogram are parallel)

From equation (1) and (2), we obtains and


AD and CF are opposite sides of quadrilateral ACFD which are both equal and parallel to each other. Thus, it is a parallelogram.

And because ACFD is a parallelogram with opposite sides both parallel and equal length.

In and ,

(Given)

(Given)

(Opposite sides of a parallelogram)

So, by SSS congruence condition.
Q12 ABCD is a trapezium in which and (see fig.). Show that




Diagonal diagonal
Solution:
Given,

Trapezium ABCD


Construction: Draw a line through C parallel to DA intersecting AB produced at E.

is given and by construction therefore,
AECD is a parallelogram therefore,

(Opposite sides of a parallelogram)

(Given)
Therefore,


…..equation (1) (Angle of opposite to equal side of a triangle are equal)

………….equation (2) (Linear pair Axiom)

……….equation (3) (The sum of consecutive interior angle on the sum side of the transversal is )
From Equation (2) and (3)


But

Or



(Angles on the same side of transversal)



But ( )

⇒ ∠D = ∠C


In ,

(Common)


(Given)

Thus, by SAS congruence condition.


Diagonal diagonal by Corresponding Parts of Congruent Triangles as