Understanding Quadrilaterals — Class 8 Mathematics
1. Polygons — The Big Picture
A POLYGON is a simple closed curve made entirely of LINE SEGMENTS. Classification by number of sides: Triangle (3). Quadrilateral (4). Pentagon (5). Hexagon (6). Heptagon (7). Octagon (8). Nonagon (9). Decagon (10). n-gon (n sides).
Types of Polygons
- Convex: ALL interior angles < 180°. All diagonals lie INSIDE. Example: regular hexagon.
- Concave: At least ONE interior angle > 180°. Some diagonals lie OUTSIDE. Example: arrowhead.
- Regular: All sides equal AND all angles equal. Example: square, equilateral triangle, regular hexagon.
- Irregular: Sides and/or angles NOT all equal.
Angle Sum Property of Polygons
The sum of the EXTERIOR angles of ANY convex polygon = 360° (always — regardless of number of sides). The sum of INTERIOR angles = (n−2) × 180°, where n = number of sides. Verification: Triangle: (3−2)×180° = 180°. Quadrilateral: (4−2)×180° = 360°. Pentagon: (5−2)×180° = 540°.
Finding Each Angle of a Regular Polygon
Each interior angle of a regular n-gon = [(n−2)×180°]/n. Each exterior angle = 360°/n. Example — Regular Octagon: Each exterior angle = 360°/8 = 45°. Each interior angle = 180°−45° = 135° (or [(8−2)×180°]/8 = 135°).
2. Quadrilaterals — Angle Sum = 360°
A quadrilateral has 4 sides, 4 vertices, 4 angles, and 2 diagonals. Angle sum = 360°. Proof: Draw a diagonal → splits the quadrilateral into TWO triangles. Each triangle sums to 180° → total = 2×180° = 360°. 'If three angles of a quadrilateral are known, the fourth = 360° − sum of the known three.'
3. Types of Quadrilaterals — The Family Tree
Trapezium
A quadrilateral with AT LEAST ONE PAIR of parallel sides. If the non-parallel sides are equal, it's an ISOSCELES TRAPEZIUM. 'A trapezium looks like a triangle with its top cut off.'
Kite
A quadrilateral with TWO PAIRS of EQUAL ADJACENT sides. Properties: ONE diagonal bisects the other. Diagonals are PERPENDICULAR. ONE pair of opposite angles are equal. 'A kite is the ONLY quadrilateral where diagonals are always perpendicular — but they don't necessarily bisect each other.'
Parallelogram
A quadrilateral with BOTH PAIRS of opposite sides PARALLEL. Properties:
- Opposite sides are EQUAL (AB = CD, BC = DA).
- Opposite angles are EQUAL (∠A = ∠C, ∠B = ∠D).
- Adjacent angles are SUPPLEMENTARY (∠A+∠B = 180°).
- Diagonals BISECT each other (AO = OC, BO = OD).
- Diagonals are NOT necessarily equal.
Rectangle
A parallelogram where EVERY angle = 90°. Inherits ALL parallelogram properties PLUS: diagonals are EQUAL. 'Every rectangle is a parallelogram. The special feature of a rectangle: diagonals =.'
Rhombus
A parallelogram where ALL FOUR SIDES are EQUAL. Inherits ALL parallelogram properties PLUS: diagonals are PERPENDICULAR BISECTORS of each other. Diagonals BISECT the angles. 'Every rhombus is a parallelogram. Every square is a rhombus. But not every rhombus is a square.'
Square
A rectangle AND a rhombus combined. ALL properties apply: all sides equal, all angles 90°, diagonals equal AND perpendicular bisectors, diagonals bisect angles. 'The square is the MOST symmetric quadrilateral. It has ALL properties.'
4. Tests for a Parallelogram (Ways to Prove)
A quadrilateral IS a parallelogram if ANY ONE of these is true:
- Both pairs of opposite sides are EQUAL.
- Both pairs of opposite angles are EQUAL.
- Diagonals BISECT each other.
- One pair of opposite sides is BOTH parallel AND equal.
5. Worked Examples
Example 1 — Finding Unknown Angles: In a quadrilateral, ∠A = 120°, ∠B = 80°, ∠C = 70°. Find ∠D. ∠D = 360° − (120°+80°+70°) = 360° − 270° = 90°.
Example 2 — Parallelogram Angles: In parallelogram ABCD, ∠A = 2x+10°, ∠B = 3x−20°. Find all angles. ∠A + ∠B = 180° (adjacent angles supplementary). (2x+10) + (3x−20) = 180 → 5x−10 = 180 → x = 38. ∠A = 86°, ∠B = 94°. ∠C = ∠A = 86°. ∠D = ∠B = 94°.
Example 3 — Quadrilateral Class Identification: A quadrilateral has all sides equal and diagonals equal. What is it? All sides equal → rhombus. Diagonals also equal → SQUARE (a rhombus with equal diagonals = square).
Example 4 — Proving a Rhombus: In a parallelogram, if one diagonal bisects an angle, prove it is a rhombus. Given: AC bisects ∠A. To Prove: ABCD is a rhombus. In ΔABC and ΔADC: AC = AC (common), ∠BAC = ∠DAC (bisector), ∠BCA = ∠DAC (alternate interior, AD∥BC). By ASA → ΔABC ≅ ΔADC → AB = AD. Since opposite sides of a parallelogram are equal, all four sides are equal → rhombus.
6. Special Properties Summary Table
| Property | ∥gram | Rect | Rhomb | Square | Kite | Trap |
|---|---|---|---|---|---|---|
| Opposite sides ∥ | ✓ | ✓ | ✓ | ✓ | ✗ | 1 pair |
| Opposite sides = | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ |
| All sides = | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ |
| All ∠s = 90° | ✗ | ✓ | ✗ | ✓ | ✗ | ✗ |
| Diagonals bisect each other | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ |
| Diagonals = | ✗ | ✓ | ✗ | ✓ | ✗ | ✗ |
| Diagonals ⟂ | ✗ | ✗ | ✓ | ✓ | ✓ | ✗ |
| Diagonals bisect ∠s | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ |
7. Common Mistakes
- 'Every parallelogram is a rectangle' — Only parallelograms with 90° angles are rectangles.
- 'Diagonals of every quadrilateral bisect each other' — Only parallelograms (and their special cases).
- Confusing rhombus and kite — Rhombus: ALL sides =. Kite: TWO PAIRS of ADJACENT sides =.
- Sum of exterior angles = 360° for ALL convex polygons — Even a 100-sided polygon!
8. AP Exam Focus
| Topic | Marks |
|---|---|
| Angle sum of polygons | 2-3 |
| Properties of quadrilaterals | 3-4 |
| Parallelogram tests | 3-4 |
| Finding unknown angles | 4-5 |
Key Exam Tips
- Draw the quadrilateral and mark ALL given information before solving.
- For 'find angle x' problems: use angle sum (360°) + properties (opposite angles equal, adjacent supplementary).
- Know the specific properties of EACH type — they build on each other. Square has ALL. Trapezium has the LEAST.
- 'Interior + Exterior = 180°' at every vertex — use this for quick angle finding in regular polygons.
Quick Self-Test
- Sum of interior angles of a hexagon? (Answer: (6−2)×180° = 720°.)
- Each exterior angle of a regular pentagon? (Answer: 360°/5 = 72°.)
- In a parallelogram, if one angle is 70°, find the other three. (Answer: 70°, 110°, 110° — opposite equal, adjacent supplementary.)
- A quadrilateral has diagonals that bisect each other and are equal. Identify it. (Answer: Rectangle.)
- Which quadrilateral has perpendicular diagonals but is NOT a rhombus? (Answer: Kite.)
