What this chapter is about
A quadrilateral is a closed figure made of four straight sides. You see quadrilaterals everywhere: the top of your desk, a kite, a cricket pitch, floor tiles, window panes. In earlier classes you learnt about triangles and their angle sum. Now you will extend that idea to four-sided figures and discover that the sum of interior angles of every quadrilateral is 360°.
This chapter introduces you to special quadrilaterals such as trapeziums, parallelograms, rhombuses, rectangles and squares. You will learn what properties each one has — which sides are equal, which angles are equal, how the diagonals behave. Understanding these properties helps you solve problems about unknown angles or sides and also prepares you for later work in geometry and mensuration.
By the end of the chapter you should be able to classify a quadrilateral by its properties, use the angle-sum property to find missing angles, and apply diagonal properties to solve simple problems.
Key ideas
- A quadrilateral has four vertices, four sides and four interior angles. The sum of its interior angles is always 360°.
- A trapezium has exactly one pair of opposite sides parallel.
- A parallelogram has both pairs of opposite sides parallel. Its opposite sides are equal, opposite angles are equal, and the diagonals bisect each other (cut each other into two equal parts).
- A rhombus is a parallelogram with all four sides equal. Its diagonals bisect each other at right angles.
- A rectangle is a parallelogram with all four angles equal to 90°. Its diagonals are equal in length and bisect each other.
- A square is both a rhombus and a rectangle: all sides equal, all angles 90°, diagonals equal and bisecting at right angles.
- A kite has two pairs of adjacent sides equal. Its diagonals are perpendicular, and one diagonal bisects the other.
Formulas and facts to remember
- Fact / Formula: Sum of angles of a quadrilateral = 360° · Meaning: Add all four interior angles; the total is always 360°.
- Fact / Formula: Opposite sides of a parallelogram are equal · Meaning: If ABCD is a parallelogram, AB = CD and AD = BC.
- Fact / Formula: Opposite angles of a parallelogram are equal · Meaning: ∠A = ∠C and ∠B = ∠D.
- Fact / Formula: Consecutive angles of a parallelogram are supplementary · Meaning: ∠A + ∠B = 180°.
- Fact / Formula: Diagonals of a parallelogram bisect each other · Meaning: They meet at a point that is the midpoint of each diagonal.
- Fact / Formula: Diagonals of a rhombus bisect at 90° · Meaning: They cross at right angles.
- Fact / Formula: Diagonals of a rectangle are equal · Meaning: Each diagonal has the same length.
Worked examples
Example 1 — Finding a missing angle
In quadrilateral PQRS, the angles at P, Q and R are 85°, 110° and 95°. Find the angle at S.
Solution: Sum of angles = 360° ∠P + ∠Q + ∠R + ∠S = 360° 85 + 110 + 95 + ∠S = 360 290 + ∠S = 360 ∠S = 360 − 290 = 70°
The angle at S is 70°.
Example 2 — Using parallelogram properties
ABCD is a parallelogram. If ∠A = 65°, find the other three angles.
Solution: Opposite angles of a parallelogram are equal, so ∠C = ∠A = 65°. Consecutive angles are supplementary, so ∠B = 180° − 65° = 115°. ∠D = ∠B = 115°.
The angles are ∠A = 65°, ∠B = 115°, ∠C = 65°, ∠D = 115°.
Example 3 — Diagonal of a rhombus
The diagonals of a rhombus KLMN meet at point O. If KM = 16 cm and LN = 12 cm, find KO and the area of triangle KOL.
Solution: Diagonals of a rhombus bisect each other. KO = KM ÷ 2 = 16 ÷ 2 = 8 cm. LO = LN ÷ 2 = 12 ÷ 2 = 6 cm.
The diagonals meet at right angles, so triangle KOL is right-angled at O. Area of triangle KOL = (1/2) × KO × LO = (1/2) × 8 × 6 = 24 cm².
Common mistakes
- Adding four angles and expecting 180° → Remember that 180° is for a triangle; a quadrilateral's angles add to 360°.
- Thinking all parallelograms have 90° angles → Only rectangles and squares have all right angles; a general parallelogram does not.
- Believing diagonals of every parallelogram are equal → Diagonals are equal only in a rectangle (and square), not in a general parallelogram or rhombus.
- Confusing bisect with perpendicular bisect → In a parallelogram diagonals bisect each other but are not necessarily perpendicular; in a rhombus they are both.
- Labelling a kite as a parallelogram → A kite has adjacent sides equal, not opposite sides; it is not a parallelogram.
Quick revision
- Angle sum of any quadrilateral = 360°.
- Parallelogram: opposite sides equal, opposite angles equal, diagonals bisect each other.
- Rhombus: all sides equal, diagonals bisect at 90°.
- Rectangle: all angles 90°, diagonals equal.
- Square: all sides equal, all angles 90°, diagonals equal and perpendicular.
- Trapezium: only one pair of parallel sides.