Class 9 Maths Chapter 8: Quadrilaterals

Quadrilaterals is Chapter 8 of NCERT Class 9 Maths. It studies the properties of a parallelogram, the conditions that make a quadrilateral a parallelogram, and the Mid-point Theorem for triangles. Watch the eight video lessons below, then use these notes, theorems and solved examples to revise.

What you will learn

Why a diagonal divides a parallelogram into two congruent triangles
The properties of the opposite sides, opposite angles and diagonals of a parallelogram
The conditions that prove a quadrilateral is a parallelogram
How the diagonals behave in a rectangle, rhombus and square
The Mid-point Theorem and its converse, with applications

Video Lessons

Quadrilaterals - Part 1

Quadrilaterals - Part 2

Quadrilaterals - Part 3

Quadrilaterals - Part 4

Quadrilaterals - Part 5

Quadrilaterals - Part 6

Quadrilaterals - Part 7

Quadrilaterals - Part 8

Key Concepts and Formulas

Parallelogram Basics

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Most results in this chapter are proved by drawing a diagonal and using congruent triangles from Chapter 7.

Angle sum of a quadrilateral = 360°
Parallelogram ABCD: ∠A + ∠B = 180°
Diagonal AC: △ABC ≅ △CDA (ASA)

The 360° angle sum comes from splitting the quadrilateral into two triangles with a diagonal.
Adjacent angles of a parallelogram are supplementary because they are co-interior angles between parallel lines.
A diagonal divides a parallelogram into two congruent triangles.

Properties of a Parallelogram

Each of these is a theorem, and each one also has a converse.

In parallelogram ABCD (diagonals meet at O):
AB = CD and AD = BC
∠A = ∠C and ∠B = ∠D
OA = OC and OB = OD

Parallelogram ABCD with equal opposite sides, opposite angles of 72° and 108° and diagonals bisecting at O, Class 9 Quadrilaterals — Sides, angles and diagonals of a parallelogram

Opposite sides are equal. Converse: if both pairs of opposite sides are equal, the quadrilateral is a parallelogram.
Opposite angles are equal. Converse: if both pairs of opposite angles are equal, it is a parallelogram.
Diagonals bisect each other. Converse: if the diagonals bisect each other, it is a parallelogram.

Conditions for a Parallelogram

To prove that a quadrilateral is a parallelogram, it is enough to show any one of the following.

Both pairs of opposite sides are parallel (the definition)
Both pairs of opposite sides are equal
Both pairs of opposite angles are equal
The diagonals bisect each other
One pair of opposite sides is both equal and parallel

Rectangle, Rhombus and Square

These are special parallelograms, so all the properties above apply to them as well.

Rectangle: AC = BD
Rhombus: AC ⊥ BD
Rhombus side² = (d₁/2)² + (d₂/2)²
Square: AC = BD and AC ⊥ BD

Rectangle: a parallelogram with one angle 90° (so every angle is 90°). Its diagonals are equal.
Rhombus: a parallelogram with all sides equal. Its diagonals bisect each other at right angles.
Square: both a rectangle and a rhombus. Its diagonals are equal and are perpendicular bisectors of each other.
A parallelogram whose diagonals are equal is a rectangle.

The Mid-point Theorem

These two results are about triangles, but they are proved using a parallelogram.

In △ABC, D = mid-point of AB, E = mid-point of AC
⇒ DE ∥ BC and DE = ½ BC
Converse: D mid-point of AB, DE ∥ BC
⇒ E is the mid-point of AC

Mid-point theorem in triangle ABC where DE joins mid-points, is parallel to BC and half its length, Class 9 Maths — DE is parallel to BC and half its length

Mid-point Theorem: the line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.
Converse: a line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.
Joining the mid-points of the sides of any quadrilateral, in order, always gives a parallelogram.
Joining the mid-points of the three sides of a triangle divides it into four congruent triangles.

Solved Examples

Example 1: In parallelogram ABCD, ∠A = 72°. Find the other angles.

∠C = ∠A = 72° (opposite angles)
∠B = 180° − 72° = 108° (adjacent angles)
∠D = ∠B = 108°
∴ ∠B = 108°, ∠C = 72°, ∠D = 108°

Example 2: Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find all its angles.

4x + 5x = 180°
9x = 180° ⇒ x = 20°
Angles: 4 × 20° = 80°, 5 × 20° = 100°
∴ Angles are 80°, 100°, 80°, 100°

Example 3: The perimeter of a parallelogram is 54 cm and one side is 15 cm. Find the adjacent side.

2(a + b) = 54 ⇒ a + b = 27
15 + b = 27
∴ b = 12 cm

Example 4: The diagonals of parallelogram PQRS meet at O, with PO = (3x + 1) cm and OR = (5x − 7) cm. Find PR.

PO = OR (diagonals bisect each other)
3x + 1 = 5x − 7
2x = 8 ⇒ x = 4
PO = 3(4) + 1 = 13 cm
PR = 2 × 13
∴ PR = 26 cm

Example 5: In △ABC, D and E are the mid-points of AB and AC, and BC = 11.6 cm. Find DE.

DE = ½ × BC (Mid-point Theorem)
DE = ½ × 11.6
∴ DE = 5.8 cm (and DE ∥ BC)

Example 6: The sides of a triangle are 8 cm, 10 cm and 12 cm. Find the perimeter of the triangle formed by joining the mid-points of its sides.

New sides = ½ of each side
= 4 cm, 5 cm, 6 cm
Perimeter = 4 + 5 + 6
∴ Perimeter = 15 cm

Example 7: The diagonals of a rhombus are 16 cm and 12 cm. Find its side.

Half diagonals = 16 ÷ 2 = 8 cm, 12 ÷ 2 = 6 cm
Diagonals meet at 90°
side = √(8² + 6²) = √(64 + 36) = √100
∴ side = 10 cm

Common Mistakes to Avoid

Assuming the diagonals of every parallelogram are equal or perpendicular. Equal diagonals belong to rectangles, and perpendicular diagonals belong to rhombuses.
For the "equal and parallel" condition, the same pair of opposite sides must be both equal and parallel. One pair equal and the other pair parallel is not enough.
Applying the Mid-point Theorem without first stating that both points are mid-points. Name the triangle and write "by Mid-point Theorem" as the reason.
Mixing up the theorem and its converse. The converse starts with a mid-point and a parallel line, and concludes that the third side is bisected.

Quick Revision

Quadrilateral angle sum = 360°
Parallelogram: ∠A + ∠B = 180°
AB = CD, AD = BC; ∠A = ∠C, ∠B = ∠D
Diagonals: OA = OC, OB = OD
Mid-point segment: DE ∥ BC, DE = ½ BC

One pair of opposite sides equal and parallel is enough to prove a parallelogram
Rectangle: equal diagonals; rhombus: perpendicular diagonals; square: both