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

  • Previous

Important properties of a parallelogram

Properties of a Parallellogram

Important properties of a parallelogram

  1. Opposite sides are congruent Equation

  2. Opposite angels are congruent Equation

  3. Consecutive angles are supplementary Equation

  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 Equation . Find all the angles of the quadrilateral..

Solution:

Suppose the measures of four angles are Equation .

  • Sum of the interior angles of the quadrilateral Equation

  • Now, Equation

  • Therefore, Angles of the quadrilateral are: Equation

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

Solution:

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, Equation

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

  • Proof,

    In Equation and Equation , Equation (Common) Equation (Opposite sides of a parallelogram are equal) Equation (Given)

  • Therefore, Equation by SSS congruence condition. Equation (by Corresponding Parts of Congruent Triangles)

  • also, Equation (Sum of the angles on the same side of the transversal) Equation

  • Thus ABCD is a rectangle.

Explore Solutions for Mathematics

Sign In