# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 6

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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

• .

• 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.

1. Again and .

Therefore, quadrilateral BEFC is a parallelogram.

1. ABED is a parallelogram therefore,

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

3. Therefore, BEFC is a parallelogram.

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

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

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

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

8. In and ,

9. (Given)

10. (Given)

11. (Opposite sides of a parallelogram)

12. 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

1. In ,

2. (Common)

3. (Given)

4. Thus, by SAS congruence condition.

5. Diagonal diagonal by Corresponding Parts of Congruent Triangles as