# 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

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

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