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

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 Equation and Equation , Equation , Equation . 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. Equation and Equation

  4. Quadrilateral ACFD is a parallelogram

  5. Equation

  6. Equation

Segment f Segment f: Segment [A, D] Segment j Segment j: Segment [A, C] Segment k Segment k: Segment [C, B] Segment l Segment l: Segment [B, A] Segment m Segment m: Segment [D, F] Segment n Segment n: Segment [F, E] Segment p Segment p: Segment [E, D] Segment g Segment g: Segment [B, E] Segment q Segment q: Segment [C, F] Point A A = (-0.32, 2.44) Point A A = (-0.32, 2.44) Point A A = (-0.32, 2.44) Point D D = (3.2, 1.34) Point D D = (3.2, 1.34) Point D D = (3.2, 1.34) Point B B = (-1.24, -0.18) Point B B = (-1.24, -0.18) Point B B = (-1.24, -0.18) Point C C = (1.58, 0.44) Point C C = (1.58, 0.44) Point C C = (1.58, 0.44) Point F Point F: Point on i Point F Point F: Point on i Point F Point F: Point on i Point E Point E: Point on h Point E Point E: Point on h Point E Point E: Point on h

Two Triangles of ABC and DEF

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

Solution:

Given,

  • Equation and Equation

  • Equation

  • Equation

  • Equation

  • Equation .

  1. Equation and Equation (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 Equation and Equation .

    Therefore, quadrilateral BEFC is a parallelogram.

  3. ABED is a parallelogram therefore,

    • Equation And Equation ……equation (1) (Opposite sides of a parallelogram are equal)

    • Therefore, BEFC is a parallelogram.

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

    • From equation (1) and (2), we obtains Equation and Equation

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

  6. In Equation and Equation ,

  • Equation (Given)

  • Equation (Given)

  • Equation (Opposite sides of a parallelogram)

  • So, Equation by SSS congruence condition.

Q-12 ABCD is a trapezium in which Equation and Equation (see fig.). Show that

  1. Equation

  2. Equation

  3. Equation

  4. Diagonal Equation diagonal Equation

Segment f Segment f: Segment [A, B] Segment h Segment h: Segment [A, D] Segment i Segment i: Segment [B, C] Segment j Segment j: Segment [C, E] Segment k Segment k: Segment [D, C] Vector u Vector u: Vector[B, E] Vector u Vector u: Vector[B, E] Point A A = (-2, 3.7) Point A A = (-2, 3.7) Point A A = (-2, 3.7) Point B B = (2.2, 3.7) Point B B = (2.2, 3.7) Point B B = (2.2, 3.7) Point D D = (-3.32, 0.72) Point D D = (-3.32, 0.72) Point D D = (-3.32, 0.72) Point C Point C: Point on g Point C Point C: Point on g Point C Point C: Point on g Point E E = (4.96, 3.7) Point E E = (4.96, 3.7) Point E E = (4.96, 3.7)

Trapezium of ABCD

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

Solution:

Given,

  • Trapezium ABCD

  • Equation

  • Equation

Construction: Draw a line through C parallel to DA intersecting AB produced at E.

  1. Equation is given and Equation by construction therefore,

    AECD is a parallelogram therefore,

    • Equation (Opposite sides of a parallelogram)

    • Equation (Given)

    Therefore,

    • Equation

    • Equation …..equation (1) (Angle of opposite to equal side of a triangle are equal)

    • Equation ………….equation (2) (Linear pair Axiom)

    • Equation ……….equation (3) (The sum of consecutive interior angle on the sum side of the transversal is Equation )

    From Equation (2) and (3)

    • Equation

    • But Equation

    • Equation Or Equation

  2. Equation

    • Equation (Angles on the same side of transversal)

    • Equation

    • Equation

    • But ( Equation )

    • ⇒ ∠D = ∠C

  3. In Equation ,

    • Equation (Common)

    • Equation

    • Equation (Given)

    • Thus, Equation by SAS congruence condition.

  4. Diagonal Equation diagonal Equation by Corresponding Parts of Congruent Triangles as Equation

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