Quadrilaterals

Overview

This chapter studies four-sided figures called quadrilaterals, their angle sum, and the special types: parallelogram, rectangle, rhombus, square, trapezium and kite. We learn the properties that distinguish each one.

Key concepts

• A quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals.
• The sum of the interior angles of any quadrilateral is 360°.
• In a parallelogram opposite sides and opposite angles are equal; diagonals bisect each other.
• A rectangle has equal diagonals; a rhombus has perpendicular diagonals; a square has both.

Important formulae

• Sum of interior angles = (n − 2) × 180°; for n = 4 this is 360°.
• Each exterior angle of a regular polygon = 360° ÷ n.
• Area of parallelogram = base × height.

Solved example

1. Three angles of a quadrilateral are 80°, 95° and 100°. Find the fourth.
2. Sum of all four angles = 360°.
3. Fourth angle = 360° − (80° + 95° + 100°) = 360° − 275° = 85°.

Important questions

1. In a parallelogram one angle is 65°. Find all the other angles.
2. Prove that the diagonals of a rectangle are equal.
3. The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find each angle.
4. State two properties that distinguish a rhombus from a rectangle.

Quick revision

All quadrilateral angles add to 360°. Learn the family tree: every square is a rectangle and a rhombus, but not the reverse. Diagonal behaviour is the quickest way to tell the special quadrilaterals apart.