Quadrilaterals
1. What is a Quadrilateral?
A quadrilateral is a CLOSED figure formed by FOUR line segments.
Parts of a quadrilateral:
- 4 vertices (corners)
- 4 sides (line segments)
- 4 interior angles
- 2 diagonals (lines joining opposite vertices)
Notation: Quadrilateral ABCD has vertices A, B, C, D in ORDER.
'Quadrilaterals are everywhere — tables, books, doors, windows. Understanding their properties helps in construction and design.'
2. Angle Sum Property
The sum of ALL interior angles of a quadrilateral is 360°.
∠A + ∠B + ∠C + ∠D = 360°
Proof: Draw a diagonal. A quadrilateral is divided into TWO triangles. Each triangle has 180°. Total = 2 × 180° = 360°.
Worked Example: Three angles of a quadrilateral are 75°, 95°, and 110°. Find the fourth angle.
Let fourth angle = x. 75° + 95° + 110° + x = 360° 280° + x = 360° x = 80°
3. Types of Quadrilaterals
Trapezium
A quadrilateral with ONE pair of parallel sides.
In trapezium ABCD (AB ∥ CD):
- AB ∥ CD (only one pair parallel)
- Adjacent interior angles on the same side of parallel lines are SUPPLEMENTARY.
- ∠A + ∠D = 180°, ∠B + ∠C = 180°
Kite
A quadrilateral with TWO pairs of ADJACENT equal sides.
- AB = AD and CB = CD
- One diagonal (longer) bisects the other
- One pair of opposite angles are EQUAL (∠B = ∠D)
- Diagonals intersect at RIGHT angles
Parallelogram
A quadrilateral with BOTH pairs of opposite sides PARALLEL. Properties covered in detail in the next chapter.
Rhombus
A parallelogram with ALL FOUR sides EQUAL.
- Diagonals bisect each other at 90°
- Diagonals bisect the interior angles
Rectangle
A parallelogram with ALL angles RIGHT (90°).
- Diagonals are EQUAL and bisect each other
- Opposite sides are equal and parallel
Square
A rectangle with ALL sides EQUAL (also a rhombus with right angles).
- All properties of rectangles and rhombuses apply
- Diagonals are EQUAL, bisect each other at 90°
4. Comparison Table
| Quadrilateral | Sides | Angles | Diagonals |
|---|---|---|---|
| Trapezium | 1 pair ∥ | Supplementary on ∥ sides | — |
| Kite | 2 pairs equal (adjacent) | 1 pair equal | — at 90° |
| Parallelogram | Opposite ∥ and equal | Opposite equal | Bisect each other |
| Rhombus | All 4 equal, opposite ∥ | Opposite equal | — at 90°, bisect angles |
| Rectangle | Opposite ∥ and equal | All 90° | Equal, bisect each other |
| Square | All 4 equal, opposite ∥ | All 90° | Equal, bisect at 90° |
5. Interrelationship
'A square is a SPECIAL type of: rectangle (all angles 90°), rhombus (all sides equal), and parallelogram (opposite sides parallel).'
Set relationships: Square ⊂ Rectangle ⊂ Parallelogram Square ⊂ Rhombus ⊂ Parallelogram
6. Worked Problems
Worked Example: Find the angles of a quadrilateral if they are in the ratio 2 : 3 : 5 : 8.
Let angles = 2x, 3x, 5x, 8x. 2x + 3x + 5x + 8x = 360° 18x = 360° x = 20°
Angles: 40°, 60°, 100°, 160°.
Worked Example: In a kite ABCD with AB = AD and CB = CD, ∠ABC = 110° and ∠ADC = 70°. Find ∠BAD.
In a kite, the angles between unequal sides are equal. ∠ABC = ∠ADC? No — these are between equal and unequal sides. The angles between the equal sides are ∠ABC and... Let us reconsider. In a kite, the angles between UNEQUAL sides are equal. Here AB = AD (equal sides), CB = CD (equal sides). So ∠ABC = ∠ADC? Actually, in kite ABCD: If AB = AD and CB = CD, then ∠ABC = ∠ADC (angles between unequal sides). Given ∠ABC = 110° and ∠ADC = 70° — this contradicts unless the labelling is different. Sum of all angles = 360°. ∠ABC + ∠ADC = 180°. So ∠BAD + ∠BCD = 180°.
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'All quadrilaterals with 4 equal sides are squares' | A rhombus also has 4 equal sides but angles ≠ 90°. A square needs ALL angles 90° |
| 'Trapezium has exactly one pair of parallel sides' | Correct — but some definitions allow both pairs to be parallel, in which case it is a parallelogram |
| 'Diagonals of a kite are equal' | Diagonals of a kite are NOT equal. They intersect at 90° |
| 'Every rectangle is a square' | FALSE. A rectangle is a square only if ALL sides are equal |
ICSE Exam Focus (4–6 marks)
- 2-mark questions: Identify quadrilaterals from properties
- 3-mark questions: Find missing angles using angle sum property
- 4-mark questions: Ratio problems or algebraic angle problems
- 6-mark questions: Proof-based problems about properties
Self-Test
Q1. Find the fourth angle of a quadrilateral if three angles are 90°, 120°, and 85°. A1. 90° + 120° + 85° = 295°. Fourth = 360° — 295° = 65°.
Q2. In a parallelogram, one angle is 70°. Find the other three angles. A2. Opposite angles are equal: other angle = 70°. Adjacent angles are supplementary: 180° — 70° = 110°. Angles: 70°, 110°, 70°, 110°.
Q3. The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find them. A3. 1x + 2x + 3x + 4x = 360° → 10x = 360° → x = 36°. Angles: 36°, 72°, 108°, 144°.
Q4. Name the quadrilateral: 'Diagonals are equal, bisect each other, but are NOT perpendicular.' A4. Rectangle. (Square also has equal diagonals that bisect, but they are also perpendicular.)
Q5. In a quadrilateral, three angles are equal and the fourth is 120°. Find each equal angle. A5. Let each equal angle = x. 3x + 120° = 360° → 3x = 240° → x = 80°.
Q6. Can a quadrilateral have angles 70°, 80°, 120°, 100°? Justify. A6. Sum = 70° + 80° + 120° + 100° = 370° > 360°. NO, a quadrilateral cannot have these angles.
