NCERT Solutions Class 9 Maths Chapter 8 — Quadrilaterals

Solution:

Step 1: Any quadrilateral, whether convex or concave, can be divided into two triangles by drawing one of its diagonals.

Step 2: Since the sum of angles in each triangle is 180°, the sum of angles in the quadrilateral will be 2 × 180° = 360°.

Step 3: This property holds true for all quadrilaterals, convex or concave.

Solution:

Step 1: One of the conditions for a quadrilateral to be a parallelogram is that its opposite sides are equal in length.

Step 2: If AB = CD and BC = DA, this condition is met.

Step 3: Therefore, Ravi's reasoning is correct; the quadrilateral ABCD must be a parallelogram.

Solution:

Step 1: The given information states that the diagonals PR and QS bisect each other at point O (PO=OR and QO=OS).

Step 2: A fundamental property of a parallelogram is that its diagonals bisect each other.

Step 3: Therefore, if the diagonals of a quadrilateral bisect each other, it must be a parallelogram. It doesn't necessarily have to be a rhombus or a rectangle unless additional conditions (like perpendicular diagonals or equal diagonals) are met.

Solution:

Step 1: In a parallelogram, opposite angles are equal. So, ∠C = ∠A = 70°.

Step 2: Also, consecutive angles are supplementary (sum to 180°). So, ∠A + ∠B = 180°.

Step 3: Substituting ∠A = 70°, we get 70° + ∠B = 180°, which means ∠B = 110°.

Step 4: Since opposite angles are equal, ∠D = ∠B = 110°.

Solution:

Step 1: The conditions for a quadrilateral to be a parallelogram include: (1) both pairs of opposite sides are equal, (2) both pairs of opposite angles are equal, (3) diagonals bisect each other, and (4) one pair of opposite sides is equal and parallel.

Step 2: If only one pair of opposite sides is equal (e.g., AB = CD), it does not guarantee that the quadrilateral is a parallelogram. It could be an isosceles trapezium or another non-parallelogram figure.

Step 3: Therefore, 'One pair of opposite sides is equal' is not a sufficient condition.