CBSE Class 9 Maths Notes: Quadrilaterals

Definitions: What is a Quadrilateral?

A quadrilateral is a closed two-dimensional shape with four sides (edges) and four vertices (corners). Some common types of quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapezoids, and kites.

Diagonal of a Parallelogram:

Core Principle: A diagonal of a parallelogram divides it into two congruent triangles. This means that the two triangles formed by the diagonal are identical in all aspects – their sides and angles are equal.

Explanation: Consider parallelogram ABCD and diagonal AC. Triangles ABC and CDA are congruent (by the Side-Angle-Side (SAS) congruence criterion). This can be proven by showing that:

• AB = CD (Opposite sides of a parallelogram are equal)
• BC = DA (Opposite sides of a parallelogram are equal)
• AC = AC (Common side)

Properties of Parallelograms: Opposite Sides, Angles

Opposite Sides: In a parallelogram, opposite sides are equal in length and parallel to each other.

Opposite Angles: In a parallelogram, opposite angles are equal in measure. For example, in parallelogram ABCD, $\angle A = \angle C$ and $\angle B = \angle D$.

Adjacent Angles: Adjacent angles (angles next to each other) in a parallelogram are supplementary, meaning their sum is 180 degrees.

Characterization of a Parallelogram

A quadrilateral can be identified as a parallelogram if it satisfies any of the following conditions:

• Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are equal in length.
• One pair of opposite sides is both parallel and equal in length.
• Opposite angles are equal.
• The diagonals bisect each other.

Diagonals of a Parallelogram

Core Principle: The diagonals of a parallelogram bisect each other. This means that they cut each other in half at their point of intersection.

Converse: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Example: If AC and BD are diagonals of parallelogram ABCD and they intersect at point O, then AO = OC and BO = OD.

Mid-point Theorem

Statement: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is equal to half the length of the third side.

Example: If D and E are the midpoints of sides AB and AC of triangle ABC respectively, then DE is parallel to BC and DE = 1/2 BC.