Quadrilaterals
Class 9 Maths • Chapter 08 • Comprehensive Guide
Why Study Quadrilaterals?
- Used in tiles, windows, doors, frames, screens
- Foundation for Areas of Parallelograms & Triangles
- Important for coordinate geometry in higher classes
- High weightage in MCQs & reasoning questions
1. Basics of Quadrilaterals
A figure formed by joining four points in an order is called a quadrilateral. It has 4
sides, 4 angles, and 4 vertices.
Angle Sum Property
The sum of the angles of a quadrilateral is 360°.
\( \angle A + \angle B + \angle C + \angle D = 360^{\circ} \)
Why is the Angle Sum = 360°?
Draw a diagonal inside the quadrilateral.
It divides the quadrilateral into two triangles.
Since sum of angles of each triangle = 180°,
Total = 180° + 180° = 360°
2. The Quad Family
Explore the different types of quadrilaterals and their unique properties.
Parallelogram
- Both pairs of opposite sides are parallel.
- Opposite sides are equal.
- Opposite angles are equal.
- Diagonals bisect each other.
Rectangle
- A parallelogram with one angle 90°.
- All properties of a parallelogram.
- Diagonals are equal.
Rhombus
- A parallelogram with all adjacent sides equal.
- Diagonals bisect each other at right angles (90°).
Square
- A rectangle with all sides equal.
- All angles are 90°.
- Diagonals are equal and bisect at 90°.
Kite
- Two pairs of adjacent sides are equal.
- Diagonals intersect at 90°.
- One diagonal bisects the other.
Important Relationship
Square ⊂ Rectangle ⊂ Parallelogram
This means a square has all properties of a rectangle and a parallelogram.
Rectangle vs Rhombus
| Rectangle |
Rhombus |
| All angles are 90° |
Angles are not necessarily 90° |
| Opposite sides equal |
All sides equal |
| Diagonals are equal |
Diagonals are perpendicular |
3. Theorem Explorer
A diagonal of a parallelogram divides it into two congruent triangles.
\( \Delta ABC \cong \Delta CDA \)
A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
Important Theorems of Parallelogram
- Opposite sides are equal
- Opposite angles are equal
- Diagonals bisect each other
- A diagonal divides it into two congruent triangles
Converse Theorems (VERY IMPORTANT):
- If diagonals bisect each other → quadrilateral is a parallelogram
- If one pair of opposite sides is equal & parallel → parallelogram
4. The Mid-Point Theorem
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half
of it.
Mid-Point Magic
Click points D and E to visualize the theorem!
Click the dots on sides AB and AC.
Proof Idea (Write in Exam)
- Join midpoints of two sides of triangle
- Form two congruent triangles
- Corresponding angles are equal
- Hence the line is parallel & half the third side
Common Mistakes to Avoid
- ❌ Diagonals of every parallelogram are NOT equal
- ❌ Every rhombus is NOT a square
- ❌ Mid-point theorem applies ONLY to triangles
- ❌ Trapezium has only ONE pair of parallel sides
Exam Smart Zone ⭐
- Angle sum of quadrilateral = 360°
- Equal diagonals ⇒ Rectangle (not always square)
- Mid-point theorem gives parallel + half
- Always state the theorem before proof
Concept Mastery Quiz
1. A quadrilateral whose diagonals bisect each other at right angles is a:
2. The sum of all angles of a quadrilateral is:
3. If diagonals of a parallelogram are equal, then it is a:
4. In the Mid-Point Theorem, the line segment joining midpoints is equal to ______ the third
side.
5. Which figure has only one pair of opposite sides parallel?