NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 1

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Important properties of a parallelogram

Properties of a Parallellogram

Important properties of a parallelogram

  1. Opposite sides are congruent (AB=DC).

  2. Opposite angels are congruent (D=B).

  3. Consecutive angles are supplementary (A+D=180°).

  4. If one angle is right, then all angles are right.

  5. The diagonals of a parallelogram bisect each other.

  6. Each diagonal of a parallelogram separates it into two congruent triangles.

Q 1. The angles of quadrilateral are in the ratio 3:5:9:13 . Find all the angles of the quadrilateral..


Suppose the measures of four angles are 3x,5x,9xand13x .

  • Sum of the interior angles of the quadrilateral =360°

  • Now, 3x+5x+9x+13x=360°30x=360°x=12°

  • Therefore, Angles of the quadrilateral are: 3x=3×12°=36°5x=5×12°=60°9x=9×12°=108°13x=13×12°=156°

Q-2 If the diagonals of a parallelogram are equal, then show that it is a rectangle.


Quadrilateral poly1 Quadrilateral poly1: Polygon A, B, C, D Segment a Segment a: Segment [A, B] of Quadrilateral poly1 Segment b Segment b: Segment [B, C] of Quadrilateral poly1 Segment c Segment c: Segment [C, D] of Quadrilateral poly1 Segment d Segment d: Segment [D, A] of Quadrilateral poly1 Segment f Segment f: Segment [A, C] Segment g Segment g: Segment [D, B] Point A A = (-2.42, 4.26) Point A A = (-2.42, 4.26) Point A A = (-2.42, 4.26) Point B B = (-2.4, 1.14) Point B B = (-2.4, 1.14) Point B B = (-2.4, 1.14) Point C C = (2.6, 1.08) Point C C = (2.6, 1.08) Point C C = (2.6, 1.08) Point D D = (2.58, 4.22) Point D D = (2.58, 4.22) Point D D = (2.58, 4.22) Point D D = (2.58, 4.22) Point D D = (2.58, 4.22)

Rectangle of ABCD

Rectangle of ABCD also AC=BD

Given, AC=BD

  • To show ABCD is a rectangle we have to prove that one of its interior angle is right angled.

  • Proof,

    In ΔABC and ΔBAD , BC=BA (Common) AC=AD (Opposite sides of a parallelogram are equal) AC=BD (Given)

  • Therefore, ΔABCΔBAD by SSS congruence condition. A=B (by Corresponding Parts of Congruent Triangles)

  • also, A+B=180° (Sum of the angles on the same side of the transversal) 2A=180°A=90°

  • Thus ABCD is a rectangle.

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