Quadrilaterals — Mathematics Class 9 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Quadrilaterals (Class 9) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Quadrilaterals (CBSE & HBSE)
Properties of parallelograms dominate the chapter. CBSE asks proofs using diagonals; HBSE often tests the mid-point theorem and parallelogram conditions. 6–8 marks.
Types of Quadrilaterals
Definitions
| Quadrilateral | Definition |
|---|---|
| Trapezium | one pair of opposite sides parallel |
| Parallelogram | both pairs of opposite sides parallel |
| Rhombus | parallelogram with all four sides equal |
| Rectangle | parallelogram with one right angle (hence all four) |
| Square | parallelogram that is both rhombus and rectangle |
| Kite | two pairs of adjacent equal sides |
Angle Sum Property
Sum of interior angles of a quadrilateral = 360° (split into two triangles via a diagonal).
Hierarchy
Square ⊂ Rectangle ⊂ Parallelogram; Square ⊂ Rhombus ⊂ Parallelogram. A square is the most special: it satisfies all parallelogram properties + all rhombus properties + all rectangle properties.
Quick Recognition
| Given | Likely type |
|---|---|
| Diagonals bisect each other | Parallelogram |
| Diagonals bisect each other at right angles | Rhombus |
| Diagonals equal & bisect each other | Rectangle |
| Diagonals equal, bisect each other at right angles | Square |
CBSE Tip: State explicitly which property is being used ("diagonals bisect at right angles") — assertion-style questions need the exact name.
Properties of a Parallelogram
Five Defining Properties
- Opposite sides are equal.
- Opposite angles are equal.
- Diagonals bisect each other.
- Consecutive angles are supplementary (sum = 180°).
- A diagonal divides the parallelogram into two congruent triangles.
Converse Tests for Parallelogram
A quadrilateral is a parallelogram if any one of the following holds:
- Opposite sides equal.
- Opposite angles equal.
- Diagonals bisect each other.
- One pair of opposite sides is both equal and parallel.
Special Parallelograms — Diagonals
| Type | Diagonals |
|---|---|
| Parallelogram | bisect each other |
| Rectangle | bisect each other and are equal |
| Rhombus | bisect each other at right angles |
| Square | equal + bisect each other at right angles |
Area Formula (Bonus)
Area of parallelogram = base × height (perpendicular distance between parallel sides).
HBSE Tip: When proving a quadrilateral is a parallelogram, choose the easiest converse test based on the given data — usually "one pair equal and parallel" or "diagonals bisect".
The Mid-Point Theorem
Mid-Point Theorem (Theorem 8.9 NCERT)
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it.
If D and E are mid-points of AB and AC of △ABC, then DE ∥ BC and DE = (1/2) BC.
Converse Theorem
The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.
Useful Corollary
In any quadrilateral ABCD, joining mid-points of consecutive sides creates a parallelogram (the Varignon parallelogram).
Application in Parallelogram Proofs
Used extensively to:
- Show that the line joining mid-points of opposite sides is parallel to the other pair.
- Prove specific lengths inside a parallelogram (e.g. diagonals halved).
CBSE Tip: When invoking the mid-point theorem, identify the triangle to which it applies. Marks are deducted if the relevant triangle is unstated.
Frequently asked questions
Are these Quadrilaterals notes free?
Yes — the Quadrilaterals notes for Mathematics (Class 9) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Quadrilaterals notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Quadrilaterals chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Quadrilaterals.