These comprehensive RBSE Class 9 Maths Notes Chapter 8 Quadrilaterals will give a brief overview of all the concepts.

## RBSE Class 9 Maths Chapter 8 Notes Quadrilaterals

1. Quadrilateral : A plane figure bounded by four line segments is called a quadrilateral. In the given figure PQRS is a quadrilateral where PQ, QR, RS and SP are four sides and ∠P, ∠Q, ∠R and ∠S are four angles.

2. A quadrilateral has four sides, four angles and four vertices.

3. Vertex: The points of intersection of sides are known as vertices. Here P, Q, R and S are four vertices of quadrilateral.

4. Diagonal: Line joining opposite vertices is called diagonal. In the figure, PR and SQ are the diagonals of quadrilateral PQRS.

5. Consecutive or Adjacent Sides : Two sides of quadrilateral are consecutive or adjacent sides, if they have a common point (vertex). In the figure :

PQ & QR and PS & SR are adjacent sides.

Similarly, SP, PQ and SR, RQ are pairs of adjacent sides.

6. Opposite sides : Two sides of a quadrilateral are opposite sides, if they have no common end-point (vertex).

7. Consecutive or Adjacent Angles : Two angles of a quadrilateral are consecutive or adjacent angles if their common arm is a side of the quadrilateral. For example ∠P and ∠Q are adjacent angles.

8. Opposite Angles : Two angles of a quadrilateral are said to be opposite angles if they do not have a common arm. For example, ∠P and ∠R are opposite angles.

9. Sum of all four angles of a quadrilateral is 4 right angles (360°).

10. Types of Quadrilaterals :

(i) Kite : A quadrilateral whose two pairs of adjacent sides are equal is known as kite. In the figure WXYZ is one such quadrilateral whose two pairs of adjacent sides i.e., WX, XY and WZ, YZ are equal.

(ii) Trapezium : A quadrilateral whose one pair of opposite sides is parallel is known as trapezium. In the figure, quadrilateral ABCD has one pair of opposite sides AB and CD to each other. ABCD is a trapezium.

(iii) Parallelogram: A quadrilateral whose two pairs of opposite ‘ sides are parallel and equal is known as parallelogram. A parallelogram is also trapezium but a trapezium is not a parallelogram. In the figure, PQRS is a parallelogram as its two pairs of opposite sides PQ, RS and PS, QR are parallel to each other,

(iv) Rectangle : A quadrilateral whose opposite sides are parallel and equal and also, each angle is 90°, is known as rectangle. In other words, if one angle of a parallelogram is 90° then it is a rectangle. In the figure, EFGH is a rectangle where EF = GH, EH = FG, EF\\GH and EH || FG. A rectangle is also a parallelogram but a parallelogram is not necessarily a rectangle.

(v) Rhombus : A special parallelogram whose each side is equal is known as rhombus. In the figure TUVW is a special parallelogram i.e. rhombus whose each side is equal.

A rhombus is a parallelogram but a parallelogram is not necessarily a rhombus.

A rhombus is a trapezium but a trapezium is not a rhombus.

(vi) Square :

A special parallelogram whose all sides are equal and each angle is 90° is known as a square. In the figure, KLMN is a special rectangle called ‘square’ and all its sides are equal.

A square is a trapezium but a trapezium is not a square.

A square is a parallelogram but a parallelogram is not necessarily a square.

A square is a rectangle but a rectangle is not necessarily a square.

A square is a rhombus but a rombus is not necessarily a square.

11. Properties of Parallelograms :

Theorem 1: The diagonal of a parallelogram divides it into two congruent triangles.

Theorem 2 : Opposite sides of a parallelogram are equal.

Theorem 3 : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogrm.

Theorem 4 : Opposite angles of a parallelogram are equal.

Theorem 5: If opposite angles of a quadrilateral are equal, then it is a parallelogram.

Theorem 6 : Diagonals of a parallelogram bisect each other.

Theorem 7: If diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Theorem 8: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

Mid-point Theorem:

Theorem 9 : A line segment joining the mid-point of any two sides of a triangle is parallel to the third side and equal to half of it.

Theorem 10 (Converse of Theorem 9): The line drawn through the mid-point of one side of a triangle,.parallel to another side bisects the third side.

Properties of Parallelograms and Rectangles etc.:

(a) In a parallelogram :

(i) Opposite sides are equal.

(ii) Opposite angles are equal.

(iii) Diagonals bisect each other.

(iv) Each diagonal bisects a parallelogram into two congruent triangles.

(b) In a rectangle :

(i) Each angle is a right angle.

(ii) Opposite sides are equal.

(iii) Diagonals are equal.

(iv) Diagonals bisect each other.

(c) In a square :

(i) All four sides are equal.

(ii) Each angle is a right angle.

(iii) Diagonals are equal.

(iv) Diagonals bisect each other at right angle.

(v) Each diagonal makes an angle of 45° with sides.

(d) In a rhombus :

(i) All four sides are equal.

(ii) Opposite angles are equal.

(iii) Diagonals are perpendicular bisectors of each other.

(iv) Diagonals are bisectors of the vertex angles.

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