# 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

Q-11 In and , , . Vertices A, B, C are joined to vertices D, E, F respectively (see figure).show that

1. Quadrilateral ABCD is a parallelogram

2. Quadrilateral BEFC is a parallelogram

3. and

4. Quadrilateral ACFD is a parallelogram

Solution:

Given,

• and

• .

1. 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.

2. Again and .

Therefore, quadrilateral BEFC is a parallelogram.

3. 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

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

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

6. In and ,

• (Given)

• (Given)

• (Opposite sides of a parallelogram)

• So, by SSS congruence condition.

Q-12 ABCD is a trapezium in which and (see fig.). Show that

1. Diagonal diagonal

Solution:

Given,

• Trapezium ABCD

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

1. 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

2. In ,

• (Common)

• (Given)

• Thus, by SAS congruence condition.

3. Diagonal diagonal by Corresponding Parts of Congruent Triangles as

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