Class 8 · Mathematics · NCERT Ganita Prakash
Chapter 4: Quadrilaterals
Complete study notes covering definitions, properties, proofs, key results, and exam-ready practice questions.
Rectangle
Square
Parallelogram
Rhombus
Kite
Trapezium
Angle Sum = 360°
Congruence Proofs
Square
Parallelogram
Rhombus
Kite
Trapezium
Angle Sum = 360°
Congruence Proofs
📚 Table of Contents
Introduction to Quadrilaterals
The word “quadrilateral” comes from Latin: quadri (four) + latus (sides). A quadrilateral is a four-sided closed figure formed by four line segments.
Definition
A quadrilateral is a polygon with exactly four sides, four vertices, and four interior angles.
A quadrilateral is a polygon with exactly four sides, four vertices, and four interior angles.
🧩 Types of Quadrilaterals
RectangleAll 4 angles = 90°; opposite sides equal & parallel.
SquareAll 4 sides equal; all 4 angles = 90°.
ParallelogramBoth pairs of opposite sides are parallel.
RhombusAll 4 sides equal; a special parallelogram.
KiteTwo pairs of adjacent (neighbouring) sides are equal.
TrapeziumExactly one pair of opposite sides is parallel.
Must Remember
Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a trapezium — but NOT the other way around!
Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a trapezium — but NOT the other way around!
Rectangle
Definition
A rectangle is a quadrilateral in which all four angles are equal to 90°.
A rectangle is a quadrilateral in which all four angles are equal to 90°.
Alternative Definition (using Diagonals)
A rectangle is a quadrilateral whose diagonals are equal in length and bisect each other.
A rectangle is a quadrilateral whose diagonals are equal in length and bisect each other.
✅ Properties of a Rectangle
- Property 1: All four angles are 90°.
- Property 2: Opposite sides are equal in length — AB = CD and BC = AD.
- Property 3: Opposite sides are parallel — AB ∥ CD and BC ∥ AD.
- Property 4: Diagonals are equal in length — AC = BD.
- Property 5: Diagonals bisect each other — OA = OC and OB = OD.
📎 Proof — Diagonals of a Rectangle are Equal (Deduction 1)
In rectangle ABCD, consider triangles △ADC and △DAB:
AB = CD (opposite sides of rectangle)
∠BAD = ∠CDA = 90° (all angles of rectangle)
AD = AD (common side)By SAS: △ADC ≅ △DAB
∴ AC = BD (corresponding parts of congruent triangles)
∠BAD = ∠CDA = 90° (all angles of rectangle)
AD = AD (common side)By SAS: △ADC ≅ △DAB
∴ AC = BD (corresponding parts of congruent triangles)
📎 Proof — Diagonals Bisect Each Other (Deduction 2)
In rectangle ABCD, diagonals meet at O. Consider triangles △AOB and △COD:
Since ∠B = 90°: ∠3 + ∠1 = 90°
In △BCD: ∠3 + ∠2 + 90° = 180° → ∠3 + ∠2 = 90°
∴ ∠1 = ∠2By AAS: △AOB ≅ △COD
∴ OA = OC and OB = OD → O is midpoint of both diagonals ✓
In △BCD: ∠3 + ∠2 + 90° = 180° → ∠3 + ∠2 = 90°
∴ ∠1 = ∠2By AAS: △AOB ≅ △COD
∴ OA = OC and OB = OD → O is midpoint of both diagonals ✓
📎 No Special Angle Between Diagonals (Deduction 3)
If the diagonals are equal and bisect each other, the shape formed is always a rectangle, regardless of the angle between the diagonals. This is because all corner angles always work out to 90°.
Diagonals equal + Bisect each other ⟹ Rectangle
Real-World Use — Carpenter’s Method
Carpenters in Europe and farmers in Mozambique use this exact property to build rectangular frames! They cross two equal wooden strips at their midpoints — the endpoints automatically form a perfect rectangle.
Carpenters in Europe and farmers in Mozambique use this exact property to build rectangular frames! They cross two equal wooden strips at their midpoints — the endpoints automatically form a perfect rectangle.
Square
Definition
A square is a quadrilateral in which all four angles are 90° and all four sides are equal in length.
A square is a quadrilateral in which all four angles are 90° and all four sides are equal in length.
✅ Properties of a Square
- Property 1: All four sides are equal.
- Property 2: Opposite sides are parallel.
- Property 3: All four angles are 90°.
- Property 4: Diagonals are equal in length and bisect each other at 90°.
- Property 5: Diagonals bisect the angles of the square — each diagonal splits a 90° corner into two 45° angles.
📎 Proof — Diagonals of Square Meet at 90° (Deduction 5)
In square ABCD, diagonals meet at O. Consider △BOA and △BOC:
OA = OC (diagonals bisect each other)
AB = BC (all sides of square equal)
OB = OB (common side)By SSS: △BOA ≅ △BOC
∴ ∠BOA = ∠BOC
But ∠BOA + ∠BOC = 180° (linear pair)
∴ ∠BOA = ∠BOC = 90° ✓
OA = OC (diagonals bisect each other)
AB = BC (all sides of square equal)
OB = OB (common side)By SSS: △BOA ≅ △BOC
∴ ∠BOA = ∠BOC
But ∠BOA + ∠BOC = 180° (linear pair)
∴ ∠BOA = ∠BOC = 90° ✓
📎 Why do diagonals bisect the angles?
In △ADC: ∠1 + ∠3 + 90° = 180°
Since AD = DC (sides of square): ∠1 = ∠3
∴ ∠1 = ∠3 = 45° — diagonal bisects the 90° angle ✓
Since AD = DC (sides of square): ∠1 = ∠3
∴ ∠1 = ∠3 = 45° — diagonal bisects the 90° angle ✓
For a square with diagonal = d: Side = d ÷ √2
Key Relationship
Every square is a rectangle (all angles 90°). Every square is a rhombus (all sides equal). But a rectangle is NOT a square unless all sides are also equal.
Every square is a rectangle (all angles 90°). Every square is a rhombus (all sides equal). But a rectangle is NOT a square unless all sides are also equal.
Angles in a Quadrilateral
Just as the angles of a triangle sum to 180°, there is a similar rule for quadrilaterals.
∠A + ∠B + ∠C + ∠D = 360° (for ANY quadrilateral)
📎 Proof
Draw diagonal BD in quadrilateral ABCD. This splits it into two triangles.
In △ABD: ∠1 + ∠2 + ∠3 = 180°
In △BCD: ∠4 + ∠5 + ∠6 = 180°Adding:
(∠1 + ∠4) + (∠3 + ∠6) + ∠2 + ∠5 = 360°
In △BCD: ∠4 + ∠5 + ∠6 = 180°Adding:
(∠1 + ∠4) + (∠3 + ∠6) + ∠2 + ∠5 = 360°
These six parts are exactly the 4 angles of the quadrilateral.
∴ ∠A + ∠B + ∠C + ∠D = 360° ✓
How to use this
If three angles are known, the fourth angle = 360° − (sum of the other three).
If three angles are known, the fourth angle = 360° − (sum of the other three).
Common Misconception
A quadrilateral CANNOT have exactly three right angles with the fourth angle being different. If three angles = 90°, the fourth must also be 90° → it would be a rectangle. (90+90+90 = 270, fourth = 90°)
A quadrilateral CANNOT have exactly three right angles with the fourth angle being different. If three angles = 90°, the fourth must also be 90° → it would be a rectangle. (90+90+90 = 270, fourth = 90°)
🔢 Quick Example
Three angles of a quadrilateral: 80°, 110°, 95°
Find the fourth angle:Fourth angle = 360° − (80° + 110° + 95°)
= 360° − 285°
= 75°
Find the fourth angle:Fourth angle = 360° − (80° + 110° + 95°)
= 360° − 285°
= 75°
Parallelogram
Definition
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Relationship
A rectangle is a special parallelogram (all angles = 90°). A rhombus is also a special parallelogram (all sides equal). A square is both!
A rectangle is a special parallelogram (all angles = 90°). A rhombus is also a special parallelogram (all sides equal). A square is both!
✅ Properties of a Parallelogram
- Property 1: Opposite sides are equal — AB = CD and BC = AD.
- Property 2: Opposite sides are parallel — AB ∥ CD and BC ∥ AD.
- Property 3: Adjacent angles add up to 180° — ∠A + ∠B = 180°, etc.
- Property 4: Opposite angles are equal — ∠A = ∠C and ∠B = ∠D.
- Property 5: Diagonals bisect each other — OA = OC and OB = OD.
📎 Proof — Opposite Angles are Equal (Deduction 6)
Let ∠A = x° in parallelogram ABCD.AB ∥ CD, AD is a transversal:
∠A + ∠D = 180° → ∠D = 180° − x°
∠A + ∠D = 180° → ∠D = 180° − x°
AD ∥ BC, AB is a transversal:
∠A + ∠B = 180° → ∠B = 180° − x°
Since ∠B + ∠C = 180°:
∠C = 180° − ∠B = 180° − (180° − x°) = x°
∴ ∠A = ∠C = x° and ∠B = ∠D = (180° − x°) ✓
📎 Proof — Opposite Sides are Equal (Deduction 7)
Draw diagonal BD. In △ABD and △CDB:∠ABD = ∠CDB (alternate angles, AB ∥ DC)
∠ADB = ∠CBD (alternate angles, AD ∥ BC)
BD = BD (common side)
∠ADB = ∠CBD (alternate angles, AD ∥ BC)
BD = BD (common side)
By AAS: △ABD ≅ △CDB
∴ AD = CB and AB = CD ✓
📎 Proof — Diagonals Bisect Each Other (Deduction 8)
In parallelogram EASY, diagonals meet at O. In △AOE and △YOS:
AE = YS (opposite sides of parallelogram)
∠OAE = ∠OYS (alternate angles, AE ∥ YS)
∠OEA = ∠OSY (alternate angles)By ASA: △AOE ≅ △YOS
∴ OA = OY and OE = OS → O bisects both diagonals ✓
∠OAE = ∠OYS (alternate angles, AE ∥ YS)
∠OEA = ∠OSY (alternate angles)By ASA: △AOE ≅ △YOS
∴ OA = OY and OE = OS → O bisects both diagonals ✓
In a parallelogram with one angle = x°: angles are x°, (180°−x°), x°, (180°−x°)
🔢 Example
Parallelogram ABCD with ∠A = 30°, AB = 4 cm, AD = 5 cm.
Find all angles and sides.Angles: ∠A = 30°, ∠B = 150°, ∠C = 30°, ∠D = 150°
Sides: AB = CD = 4 cm, AD = BC = 5 cm
Find all angles and sides.Angles: ∠A = 30°, ∠B = 150°, ∠C = 30°, ∠D = 150°
Sides: AB = CD = 4 cm, AD = BC = 5 cm
Rhombus
Definition
A rhombus is a quadrilateral in which all four sides have the same length.
A rhombus is a quadrilateral in which all four sides have the same length.
Since the opposite sides of a rhombus are parallel (proved using equal alternate angles), every rhombus is also a parallelogram. All properties of a parallelogram apply!
✅ Properties of a Rhombus
- Property 1: All four sides are equal.
- Property 2: Opposite sides are parallel.
- Property 3: Adjacent angles add to 180°; opposite angles are equal.
- Property 4: Diagonals bisect each other.
- Property 5: Diagonals bisect the angles of the rhombus.
- Property 6: Diagonals intersect at 90° — they are perpendicular to each other.
📎 Proof — Diagonals of Rhombus are Perpendicular (Deduction 10)
In rhombus GAME, diagonals meet at O. In △GEO and △MEO:GE = ME (all sides of rhombus equal)
OE = OE (common)
GO = MO (diagonals bisect each other)
OE = OE (common)
GO = MO (diagonals bisect each other)
By SSS: △GEO ≅ △MEO
∴ ∠GOE = ∠MOE
But ∠GOE + ∠MOE = 180° (linear pair)
∴ ∠GOE = ∠MOE = 90° ✓
🔢 Example — Finding angles of a Rhombus
Rhombus ABCD with ∠A = 50°. Find all angles.Since opposite angles are equal: ∠C = 50°
Adjacent angles add to 180°: ∠B = ∠D = 180° − 50° = 130°
Adjacent angles add to 180°: ∠B = ∠D = 180° − 50° = 130°
Angles: 50°, 130°, 50°, 130°
Using diagonal: let diagonal bisect ∠A into two equal parts of ‘a’ each.
In △ADB: a + a + 50° = 180° → 2a = 130° → a = 65°
Important Difference
Diagonals of a rhombus are perpendicular — but they are NOT equal (unlike a square). Equal diagonals + perpendicular = square only.
Diagonals of a rhombus are perpendicular — but they are NOT equal (unlike a square). Equal diagonals + perpendicular = square only.
Kite
Definition
A kite is a quadrilateral (labelled ABCD) such that AB = BC and CD = DA. It has two pairs of equal adjacent (neighbouring) sides.
A kite is a quadrilateral (labelled ABCD) such that AB = BC and CD = DA. It has two pairs of equal adjacent (neighbouring) sides.
✅ Properties of a Kite
- Property 1: Two pairs of adjacent sides are equal: AB = BC and CD = DA.
- Property 2(i): The main diagonal BD bisects angles ∠ABC and ∠ADC.
- Property 2(ii): The diagonal BD is the perpendicular bisector of diagonal AC: AO = OC and BD ⊥ AC.
📎 Proof — BD ⊥ AC and AO = OC
In △AOB and △COB:
AB = CB (given — kite property)
OB = OB (common side)
∠ABO = ∠CBO (BD bisects ∠B — can be proved by congruence of △ABD and △CBD)By SAS: △AOB ≅ △COB
∴ AO = CO → BD bisects AC ✓
∠AOB = ∠COB = 90° → BD ⊥ AC ✓
AB = CB (given — kite property)
OB = OB (common side)
∠ABO = ∠CBO (BD bisects ∠B — can be proved by congruence of △ABD and △CBD)By SAS: △AOB ≅ △COB
∴ AO = CO → BD bisects AC ✓
∠AOB = ∠COB = 90° → BD ⊥ AC ✓
Remember: Kite vs Rhombus
In a kite, equal sides are adjacent. In a rhombus, all four sides are equal — so a rhombus is a special kite. A square is both!
In a kite, equal sides are adjacent. In a rhombus, all four sides are equal — so a rhombus is a special kite. A square is both!
🪁
Why is it called a Kite?The shape resembles the toy kite we fly — wide on top, narrowing to a point at the bottom. The diagonals form the two sticks of the kite that cross each other!
Trapezium
Definition
A trapezium is a quadrilateral with at least one pair of parallel opposite sides. The parallel sides are called the bases.
A trapezium is a quadrilateral with at least one pair of parallel opposite sides. The parallel sides are called the bases.
✅ Properties of a Trapezium
- At least one pair of opposite sides is parallel.
- Co-interior angles (between the parallel sides on the same side) add up to 180°: ∠S + ∠P = 180° and ∠R + ∠Q = 180°.
🔷 Isosceles Trapezium
Definition
An isosceles trapezium is a trapezium where the two non-parallel sides (legs) are equal in length.
An isosceles trapezium is a trapezium where the two non-parallel sides (legs) are equal in length.
✅ Properties of an Isosceles Trapezium
- Non-parallel sides (legs) are equal.
- Base angles are equal — angles at each base are equal to each other.
📎 Proof — Base Angles are Equal
In isosceles trapezium UVWX with UV ∥ XW and UX = VW.
Draw perpendiculars XY and WZ from X, W to base UV.Since XW ∥ UV:
∠XYU = ∠WZV = 90° → XWZY is a rectangle → XY = WZ
Draw perpendiculars XY and WZ from X, W to base UV.Since XW ∥ UV:
∠XYU = ∠WZV = 90° → XWZY is a rectangle → XY = WZ
In △UXY and △VWZ:
UX = VW (given)
XY = WZ (proved above)
∠XYU = ∠WZV = 90°
By RHS: △UXY ≅ △VWZ
∴ ∠U = ∠V (base angles are equal) ✓
Trapezium vs Parallelogram
A trapezium has at least one pair of parallel sides. A parallelogram has two pairs. So every parallelogram is a trapezium, but not every trapezium is a parallelogram.
A trapezium has at least one pair of parallel sides. A parallelogram has two pairs. So every parallelogram is a trapezium, but not every trapezium is a parallelogram.
Quick Comparison Table
| Property | Rectangle | Square | Parallelogram | Rhombus | Kite | Trapezium |
|---|---|---|---|---|---|---|
| All sides equal | ✗ | ✓ | ✗ | ✓ | ✗ | ✗ |
| All angles = 90° | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ |
| Opposite sides equal | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ |
| Opposite sides parallel | ✓ | ✓ | ✓ | ✓ | ✗ | One pair |
| Diagonals equal | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ |
| Diagonals bisect each other | ✓ | ✓ | ✓ | ✓ | One only | ✗ |
| Diagonals perpendicular | ✗ | ✓ | ✗ | ✓ | ✓ | ✗ |
| Diagonals bisect angles | ✗ | ✓ | ✗ | ✓ | One diagonal | ✗ |
| Opposite angles equal | ✓ (all 90°) | ✓ (all 90°) | ✓ | ✓ | One pair | ✗ |
🔗 Hierarchy — Subset Relationships
Remember This Chain
Square ⊂ Rectangle ⊂ Parallelogram ⊂ Trapezium ⊂ Quadrilateral
Square ⊂ Rectangle ⊂ Parallelogram ⊂ Trapezium ⊂ Quadrilateral
Square ⊂ Rhombus ⊂ Parallelogram ⊂ Trapezium ⊂ Quadrilateral
Kite overlaps with Rhombus (a rhombus is a special kite where all 4 sides are equal)
Chapter Summary
Rectangle
Quadrilateral with all angles = 90°. Diagonals equal and bisect each other.
Quadrilateral with all angles = 90°. Diagonals equal and bisect each other.
Square
All angles = 90° and all sides equal. Diagonals equal, perpendicular, bisect each other and bisect the angles.
All angles = 90° and all sides equal. Diagonals equal, perpendicular, bisect each other and bisect the angles.
Parallelogram
Both pairs of opposite sides parallel. Diagonals bisect each other (not necessarily equal).
Both pairs of opposite sides parallel. Diagonals bisect each other (not necessarily equal).
Rhombus
All four sides equal. Diagonals bisect each other at 90° and bisect the angles.
All four sides equal. Diagonals bisect each other at 90° and bisect the angles.
Kite
Two pairs of equal adjacent sides. One diagonal perpendicularly bisects the other.
Two pairs of equal adjacent sides. One diagonal perpendicularly bisects the other.
Trapezium
At least one pair of parallel sides. Co-interior angles add up to 180°.
At least one pair of parallel sides. Co-interior angles add up to 180°.
🔑 The sum of all interior angles in ANY quadrilateral = 360°
📌 Key Diagonal Identifiers
Diagonals equal + bisect each other→ Rectangle
Diagonals perpendicular + bisect each other→ Rhombus
Diagonals equal + perpendicular + bisect each other→ Square
Diagonals bisect each other (not equal, not perp.)→ Parallelogram
One diagonal ⊥ bisects other→ Kite
No special diagonal property→ Trapezium / General quad.
Exam Practice Questions
🟢 1-Mark Questions
Q1. What is the sum of all interior angles of a quadrilateral?
Ans: 360°
Q2. The diagonals of a rhombus intersect at what angle?
Ans: 90° (they are perpendicular to each other)
Q3. Is every square a rectangle? Is every rectangle a square?
Ans: Yes, every square is a rectangle. No, every rectangle is NOT a square (unless all sides are equal).
Q4. How many pairs of parallel sides does a trapezium have?
Ans: At least one pair (exactly one pair in a non-parallelogram trapezium).
🟡 2-Mark Questions
Q5. Three angles of a quadrilateral are 80°, 95°, and 110°. Find the fourth angle.
Ans: Fourth angle = 360° − (80° + 95° + 110°) = 360° − 285° = 75°
Q6. In a parallelogram ABCD, ∠A = 65°. Find all the remaining angles.
Ans: ∠A = 65°, ∠B = 115°, ∠C = 65°, ∠D = 115° (opposite angles equal; adjacent angles add to 180°)
Q7. The angles of a quadrilateral are in ratio 1 : 2 : 3 : 4. Find all angles.
Ans: Total parts = 10. Each part = 360°/10 = 36°. Angles: 36°, 72°, 108°, 144°
Q8. In a rhombus ABCD, one angle is 60°. Find all angles and the angles made by the diagonals at the sides.
Ans: ∠A = ∠C = 60°; ∠B = ∠D = 120°. Each diagonal bisects the angle, so it makes 30° (half of 60°) and 60° (half of 120°) with the sides.
🔴 3-Mark / Proof Questions
Q9. Prove that the diagonals of a rectangle are equal.
Ans: In rectangle ABCD, consider △ADC and △DAB. AB = DC (opposite sides), ∠BAD = ∠CDA = 90°, AD = AD (common). By SAS, △ADC ≅ △DAB. ∴ AC = BD (CPCT). ✓
Q10. State whether true or false and justify: “A quadrilateral whose diagonals bisect each other must be a parallelogram.”
Ans: TRUE. If diagonals bisect each other at O, then in △AOB and △COD: AO = CO, OB = OD (given), ∠AOB = ∠COD (vertically opposite). By SAS, △AOB ≅ △COD. So AB = CD and ∠OAB = ∠OCD (alternate angles) → AB ∥ CD. Similarly AD ∥ BC. Hence it is a parallelogram. ✓
Q11. PAIR and RODS are two rectangles sharing side RI. If ∠RIA = 30°, find ∠IOD.
Ans: In rectangle PAIR, diagonal RI makes ∠RIA = 30° with IA. Since PAIR is a rectangle, ∠AIR and its supplementary angles help us determine ∠RIO. ∠AIP = 90° (rectangle angle). ∠IOD = 90° − 30° = 60°. (Using angle sum in the figure: ∠IOD = 60°)
Top Exam Tips
1. Always write the angle sum property clearly when finding angles. 2. For congruence proofs: identify common sides, alternate angles, and vertically opposite angles first. 3. Quote the congruence condition (SAS, AAS, SSS, ASA, RHS) explicitly. 4. Remember: diagonals of a parallelogram are NOT equal (unless it’s a rectangle).
1. Always write the angle sum property clearly when finding angles. 2. For congruence proofs: identify common sides, alternate angles, and vertically opposite angles first. 3. Quote the congruence condition (SAS, AAS, SSS, ASA, RHS) explicitly. 4. Remember: diagonals of a parallelogram are NOT equal (unless it’s a rectangle).
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