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

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An Aritistic Rendering of Types of Quadileterals Which Also …

A parallelogram is a quadrilateral with opposite sides parallel

Parallelogram of ABCD Its Opposite Side Are Parallel

Trapezium is a quadrilateral with one pair of sides parallel

Trapeziumtrapezium

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
Two Triangles of ABC and DEF

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
Trapezium of ABCD

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