1. Chapter at a Glance
Doors, books, kites, tiles, cricket pitches and table tops are all quadrilaterals — closed figures with four sides. Some have parallel sides, some have equal sides, some have right angles, and their diagonals behave in special ways. In this chapter you will learn the family of quadrilaterals, their defining properties, the angle sum of 360°, and how to use these properties to find unknown sides and angles.
- Parts of a quadrilateral and the angle sum property.
- Parallelogram, rectangle, rhombus, square, trapezium and kite.
- Properties of their sides, angles and diagonals.
- The family tree (hierarchy) of quadrilaterals.
- Solving problems using these properties.
2. Quadrilateral Basics
A quadrilateral ABCD has four sides (AB, BC, CD, DA), four angles, four vertices and two diagonals (AC and BD). Sides that share a vertex are adjacent; sides that do not meet are opposite.
- A quadrilateral is convex if both diagonals lie inside it; otherwise it is concave.
The four angles of any quadrilateral add up to 360°. Reason: a diagonal splits it into two triangles, each with angle sum 180°.
Three angles of a quadrilateral are 75°, 90° and 115°. Find the fourth.
Solution: 360° − (75° + 90° + 115°) = 360° − 280° = 80°.
3. Types of Quadrilaterals
| Type | Definition | Key properties |
|---|---|---|
| Trapezium | At least one pair of parallel sides | Co-interior angles between parallel sides add to 180° |
| Kite | Two pairs of equal adjacent sides | Diagonals perpendicular; one diagonal bisects the other |
| Parallelogram | Both pairs of opposite sides parallel | Opposite sides equal; opposite angles equal; adjacent angles add to 180°; diagonals bisect each other |
| Rectangle | Parallelogram with a right angle | All angles 90°; diagonals equal and bisect each other |
| Rhombus | Parallelogram with all sides equal | Diagonals bisect each other at 90°; diagonals bisect the angles |
| Square | Rectangle with all sides equal | All sides equal, all angles 90°; diagonals equal, bisect each other at 90° |
4. Properties of a Parallelogram — with Reasons
- Opposite sides are equal: the diagonal AC divides ABCD into ΔABC and ΔCDA, which are congruent (ASA, using alternate angles and the common side AC). So AB = CD and BC = DA.
- Opposite angles are equal: from the same congruence, ∠B = ∠D; similarly ∠A = ∠C.
- Adjacent angles add to 180°: AB ∥ DC with AD as a transversal makes ∠A and ∠D co-interior angles.
- Diagonals bisect each other: ΔAOB ≅ ΔCOD (ASA), so AO = OC and BO = OD.
In parallelogram PQRS, ∠P = (3x + 10)° and ∠Q = (2x + 20)°. Find all angles.
Solution: Adjacent angles add to 180°: 5x + 30 = 180 → x = 30. ∠P = 100°, ∠Q = 80°, ∠R = 100°, ∠S = 80°.
5. The Family Tree
Every square is a rectangle and a rhombus; every rectangle and rhombus is a parallelogram; every parallelogram is a trapezium (it has at least one pair of parallel sides). A square is also a kite. So a property true for parallelograms is automatically true for rectangles, rhombuses and squares.
| Property | Parallelogram | Rectangle | Rhombus | Square |
|---|---|---|---|---|
| Opposite sides parallel | ✔ | ✔ | ✔ | ✔ |
| All sides equal | ✘ | ✘ | ✔ | ✔ |
| All angles 90° | ✘ | ✔ | ✘ | ✔ |
| Diagonals bisect each other | ✔ | ✔ | ✔ | ✔ |
| Diagonals equal | ✘ | ✔ | ✘ | ✔ |
| Diagonals perpendicular | ✘ | ✘ | ✔ | ✔ |
6. Solved Problems (NCERT Style)
The diagonals of a rhombus are 16 cm and 12 cm. Find its side.
Solution: The diagonals bisect each other at 90°, forming right triangles with legs 8 cm and 6 cm. By the Baudhāyana–Pythagoras theorem (Part II), side = √(64 + 36) = 10 cm.
The diagonals of rectangle ABCD meet at O. If OA = 2x + 4 and OD = 3x + 1, find x and the diagonal.
Solution: Diagonals are equal and bisect each other, so OA = OD: 2x + 4 = 3x + 1 → x = 3. OA = 10, so the diagonal = 20 units.
In trapezium ABCD with AB ∥ DC, ∠A = 65° and ∠B = 110°. Find ∠D and ∠C.
Solution: AD is a transversal: ∠A + ∠D = 180° → ∠D = 115°. BC is a transversal: ∠B + ∠C = 180° → ∠C = 70°.
7. Master Formula Table
| Fact | Rule |
|---|---|
| Angle sum of quadrilateral | 360° |
| Parallelogram adjacent angles | sum 180° |
| Parallelogram opposite angles/sides | equal |
| Parallelogram diagonals | bisect each other |
| Rectangle diagonals | equal + bisect |
| Rhombus diagonals | ⊥ bisectors; side² = (d₁/2)² + (d₂/2)² |
| Square diagonal | equal, ⊥, bisect; diagonal = side × √2 |
| Kite diagonals | ⊥; one bisects the other |
8. Master Summary Table
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Trapezium | 1 pair parallel | Co-interior sum 180° | No special rule |
| Kite | 2 pairs equal adjacent | One pair of opposite angles equal | ⊥ |
| Parallelogram | Opposite equal and parallel | Opposite equal | Bisect each other |
| Rectangle | Opposite equal | All 90° | Equal, bisect |
| Rhombus | All equal | Opposite equal | ⊥ bisectors |
| Square | All equal | All 90° | Equal, ⊥ bisectors |
9. Mind Map
- Sides, angles, diagonals
- Angle sum 360°
- Rectangle
- Rhombus
- Square
- Trapezium
- Kite
- Bisect, equal
- Perpendicular
10. Tips and Tricks
- Draw the diagonals — most problems become triangle problems.
- In parallelograms, "adjacent 180°, opposite equal" solves most angle questions.
- For a rhombus, use half-diagonals as legs of a right triangle.
- Thinking the diagonals of every parallelogram are equal (only for rectangles and squares).
- Saying a rhombus has all angles 90° (only a square does).
- Forgetting that a square is also a rhombus and a rectangle.
11. Important One-liners
- The angles of a quadrilateral add up to 360°.
- Opposite sides and opposite angles of a parallelogram are equal.
- Adjacent angles of a parallelogram are supplementary.
- The diagonals of a parallelogram bisect each other.
- Rectangle diagonals are equal.
- Rhombus diagonals are perpendicular bisectors of each other.
- A square is both a rectangle and a rhombus.
- A kite's diagonals are perpendicular.
12. Exam-pattern Questions with Detailed Answers
Which quadrilateral has diagonals that are equal and perpendicular?
- Rectangle
- Rhombus
- Square
- Kite
Show answer
Rectangle diagonals are equal but not ⊥; rhombus diagonals are ⊥ but not equal; a square has both.
Answer: (C) Square
One angle of a parallelogram is 70°. Find the others.
Show answer
Adjacent angle = 110°; opposite angles are equal: 70°, 110°, 70°, 110°.
Answer: 110°, 70°, 110°
The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find them.
Show answer
10x = 360° → x = 36°. Angles 36°, 72°, 108°, 144°.
Answer: 36°, 72°, 108°, 144°
The perimeter of a parallelogram is 50 cm and one side is 5 cm longer than the other. Find the sides.
Show answer
2(x + x + 5) = 50 → 4x + 10 = 50 → x = 10. Sides 10 cm and 15 cm.
Answer: 10 cm and 15 cm
In rhombus ABCD, ∠ABC = 120°. Find ∠ACB.
Show answer
∠BCD = 180° − 120° = 60°. The diagonal AC bisects ∠C, so ∠ACB = 30°.
Answer: 30°
Which statement is FALSE?
- Every square is a rhombus
- Every rectangle is a parallelogram
- Every rhombus is a square
- Every parallelogram is a trapezium
Show answer
A rhombus need not have right angles, so it need not be a square.
Answer: (C)
The diagonals of a parallelogram are 18 cm and 24 cm, and they meet at O. Find OA and OB.
Show answer
Diagonals bisect each other: OA = 9 cm, OB = 12 cm.
Answer: 9 cm and 12 cm
A kite has angles 110°, 110°, x and 2x (the equal pair between unequal sides). Find x.
Show answer
110 + 110 + x + 2x = 360 → 3x = 140 → x = 46⅔°.
Answer: x = 46⅔°
The side of a rhombus is 13 cm and one diagonal is 24 cm. Find the other diagonal.
Show answer
Half of the known diagonal = 12. Other half-diagonal = √(13² − 12²) = √25 = 5. Other diagonal = 10 cm.
Answer: 10 cm
Show that if the diagonals of a parallelogram are equal, it is a rectangle.
Show answer
In parallelogram ABCD with AC = BD: ΔABC ≅ ΔDCB by SSS (AB = DC, BC common, AC = DB). So ∠ABC = ∠DCB. These are adjacent angles adding to 180°, so each is 90°, making ABCD a rectangle.
Answer: Each angle becomes 90°
In parallelogram ABCD, the bisectors of ∠A and ∠B meet at P. Find ∠APB.
Show answer
∠A + ∠B = 180°, so half of each adds to 90°. In ΔAPB, ∠APB = 180° − 90° = 90°.
Answer: 90°
A square has a diagonal of 10 cm. Find its area.
Show answer
One diagonal splits the square into two triangles, each with base 10 cm (the diagonal) and height 5 cm (half the other diagonal, since the diagonals bisect each other at 90°). Area = 2 × ½ × 10 × 5 = 50 cm².
Answer: 50 cm²
13. Quick Revision Summary
- Quadrilateral angle sum = 360°.
- Parallelogram: opposite sides/angles equal, adjacent angles 180°, diagonals bisect.
- Rectangle: + equal diagonals; rhombus: + ⊥ diagonals; square: both.
- Trapezium: one pair parallel; kite: two pairs equal adjacent sides, ⊥ diagonals.
- Use triangles (via diagonals) to prove and calculate.