# NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 6 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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

- 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.

Q-12 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