Special Types of Quadrilaterals
1. Properties of a Parallelogram
A parallelogram has BOTH pairs of opposite sides PARALLEL. Key properties:
| Property | Statement | Visual |
|---|---|---|
| 1 | Opposite sides are EQUAL | AB = CD, AD = BC |
| 2 | Opposite angles are EQUAL | ∠A = ∠C, ∠B = ∠D |
| 3 | Adjacent angles are SUPPLEMENTARY | ∠A + ∠B = 180°, ∠B + ∠C = 180° |
| 4 | Diagonals BISECT each other | AO = OC, BO = OD (O is intersection) |
| 5 | Each diagonal DIVIDES the parallelogram into TWO CONGRUENT triangles | ∆ABC ≅ ∆CDA |
Worked Example: In a parallelogram ABCD, ∠A = 65°. Find the other angles.
∠C = ∠A = 65° (opposite angles equal) ∠B = 180° — 65° = 115° (adjacent supplementary) ∠D = ∠B = 115° (opposite angles equal)
2. Types of Parallelograms
Rectangle
- A parallelogram with ALL angles 90°
- Diagonals are EQUAL (AC = BD)
Rhombus
- A parallelogram with ALL sides EQUAL
- Diagonals are PERPENDICULAR (intersect at 90°)
- Diagonals BISECT the interior angles
Square
- A parallelogram with ALL sides EQUAL AND all angles 90°
- Diagonals are EQUAL, PERPENDICULAR, and BISECT angles
3. Tests for a Parallelogram
A quadrilateral ABCD is a parallelogram if ANY ONE of these conditions is true:
| Test | Condition |
|---|---|
| 1 | BOTH pairs of opposite sides are PARALLEL |
| 2 | BOTH pairs of opposite sides are EQUAL |
| 3 | BOTH pairs of opposite angles are EQUAL |
| 4 | ONE pair of opposite sides is BOTH parallel AND equal |
| 5 | Diagonals BISECT each other |
'These tests are useful for PROVING that a quadrilateral is a parallelogram without drawing all sides parallel.'
4. Midpoint Theorem
Statement: The line segment joining the MIDPOINTS of any two sides of a triangle is PARALLEL to the third side and HALF its length.
In triangle ABC:
- D is the midpoint of AB
- E is the midpoint of AC
- Then DE ∥ BC and DE = ½ BC
Proof:
- In ∆ABC, D and E are midpoints of AB and AC.
- Extend DE to F such that EF = DE.
- Join CF.
- In ∆ADE and ∆CFE: AE = EC (given), DE = EF (construction), ∠AED = ∠CEF (vertically opposite).
- ∆ADE ≅ ∆CFE (SAS). So AD = CF and ∠ADE = ∠CFE.
- Since AD = DB and AD = CF, DB = CF.
- Since ∠ADE = ∠CFE (alternate angles), AD ∥ CF, so DB ∥ CF.
- DBCF is a parallelogram. Hence DF ∥ BC and DF = BC.
- Since DE = ½ DF, we get DE ∥ BC and DE = ½ BC.
5. Converse of Midpoint Theorem
Statement: The line drawn through the MIDPOINT of one side of a triangle PARALLEL to another side BISECTS the third side.
In triangle ABC: D is midpoint of AB and DE ∥ BC. Then E is the midpoint of AC.
6. Proofs in Quadrilaterals
Worked Example: Prove that the diagonals of a rectangle are equal.
Proof: In rectangle ABCD, consider ∆ABC and ∆DCB. AB = DC (opposite sides of rectangle) BC = BC (common) ∠ABC = ∠DCB = 90° (angles of rectangle) ∆ABC ≅ ∆DCB (SAS) Therefore AC = DB (corresponding parts of congruent triangles).
Worked Example: Prove that the diagonals of a rhombus are perpendicular.
Proof: In rhombus ABCD, let diagonals intersect at O. AB = BC = CD = DA (sides of rhombus) AO = OC, BO = OD (diagonals bisect) Consider ∆AOB and ∆COB: AO = OC, AB = BC, OB = OB. ∆AOB ≅ ∆COB (SSS) ∠AOB = ∠COB (CPCT) Since ∠AOB + ∠COB = 180° (linear pair), ∠AOB = ∠COB = 90°. Hence diagonals are perpendicular.
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Midpoint theorem says the segment is equal to the side' | It says PARALLEL and HALF. Not equal. |
| 'Only one pair of opposite sides equal → parallelogram' | Need BOTH pairs equal OR one pair both parallel AND equal |
| 'Diagonals of a parallelogram are equal' | Only rectangles and squares have EQUAL diagonals |
| 'Rhombus diagonals are equal' | Rhombus diagonals are PERPENDICULAR, not equal |
ICSE Exam Focus (6–8 marks)
- 2-mark questions: Identify properties of special quadrilaterals
- 4-mark questions: Apply midpoint theorem to find lengths
- 6-mark questions: Prove a quadrilateral is a parallelogram using tests
- 8-mark questions: Multi-step proofs involving midpoint theorem and properties
Self-Test
Q1. In parallelogram ABCD, AB = 8 cm, BC = 6 cm, and ∠A = 70°. Find CD, AD, and ∠C. A1. CD = AB = 8 cm, AD = BC = 6 cm, ∠C = ∠A = 70°.
Q2. In a triangle ABC, D and E are midpoints of AB and AC. If BC = 14 cm, find DE. A2. DE = ½ BC = 7 cm.
Q3. Prove that a quadrilateral with vertices A(0,0), B(4,0), C(6,3), D(2,3) is a parallelogram. A3. AB = 4, DC = 4. AB ∥ DC (both horizontal). AD: slope = 3/2, BC: slope = 3/2. AD ∥ BC. Both pairs opposite sides parallel → parallelogram.
Q4. In ∆ABC, D is the midpoint of AB and E is the midpoint of AC. If DE = 5 cm, find BC. A4. DE = ½ BC → BC = 2 × DE = 2 × 5 = 10 cm.
Q5. Show that the diagonals of a square are equal and perpendicular. A5. Equal: Same proof as rectangle (all angles 90°). Perpendicular: Same proof as rhombus (all sides equal). Since a square is both a rectangle and a rhombus, both properties hold.
Q6. In a parallelogram, one side is 5 cm and the perimeter is 26 cm. Find the adjacent side. A6. Perimeter = 2(l + b) = 26. l + b = 13. l = 5, so b = 13 — 5 = 8 cm.
