Class 9 Maths – Quadrilaterals

8.1 Properties of a Parallelogram

• A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

Properties of a Parallelogram:

1. Opposite sides are equal: AB = CD, BC = AD
2. Opposite angles are equal: ∠A = ∠C, ∠B = ∠D
3. Diagonals bisect each other
4. Sum of adjacent angles = 180°
5. Each diagonal divides the parallelogram into two congruent triangles

Example:

• ABCD is a parallelogram → AB || CD, AD || BC

Special types of parallelogram:

• Rectangle: All angles 90°
• Square: All sides equal + all angles 90°
• Rhombus: All sides equal

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8.2 The Mid-point Theorem

• Statement: The line joining the midpoints of two sides of a triangle is parallel to the third side and half of its length.

Mathematical Representation:

• In ΔABC, D and E are midpoints of AB and AC

$$DE || BC \quad \text{and} \quad DE = \frac{1}{2} BC$$DE∣∣BCandDE=21​BC

Proof (Simple Idea):

• Using properties of parallel lines and triangles, DE || BC and DE = ½ BC.

Application:

• Used in geometry problems, coordinate geometry, and proving properties of quadrilaterals.

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Quick Short Q&A (Most Possible)

Question	Short Answer
Definition of parallelogram?	Quadrilateral with opposite sides parallel
Opposite sides of parallelogram?	Equal
Opposite angles of parallelogram?	Equal
Diagonals of parallelogram?	Bisect each other
Adjacent angles sum?	180°
Special types of parallelogram?	Rectangle, Rhombus, Square
Midpoint theorem statement?	Line joining midpoints of two sides
Midpoints of triangle sides?	DE
Diagonal divides parallelogram?	Into two congruent triangles
Uses of midpoint theorem?	Geometry and coordinate proofs