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

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An aritistic rendering of types of quadileterals which also dipicts the correct classification

Types of Quadileterals

An aritistic rendering of types of quadileterals which also dipicts the correct classification

A parallelogram is a quadrilateral with opposite sides parallel

Parallelogram of ABCD its opposite side are parallel

Parallelogram of ABCD

Parallelogram of ABCD its opposite side are parallel

Trapezium is a quadrilateral with one pair of sides parallel

Image of trapeziumImage of trapezium

Image of Trapezium

Image of trapeziumImage of trapezium

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

Two triangle of ABC and DEF, AB=DE, AB||DE, BC=EF and BC||EF

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

Trapezium of ABCD

Trapezium of ABCD, also AB∥CD and AD=BC

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