# NCERT Class 9 Mathematics Quadrilaterals CBSE Board Sample Problems (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Quadrilateral**

A quadrilateral is the union of four line-segments determined by four distinct coplanar points of which no three are collinear and the line-segments intersect only at end points.

For ABCD to be quadrilateral, following condition are required

a) The four points A, B, C and D must be distinct and co-planar.

b) No three of points A, B, C and D are co-linear.

C) Line segments i.e.. AB, BC, CD, DA intersect at their end points only.

**A quadrilateral is a four-sided polygon with four angles. There are many kinds of quadrilaterals. The five most common types are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus**.

**Angle Property of Quadrilateral**

1) Sum of all the interior angles is

2) Sum of all the exterior angles is

**Parallelogram**

A quadrilateral which has both pairs of opposite sides parallel is called a parallelogram.

Its properties are:

- The opposite sides of a parallelogram are equal.
- The opposite angles of a parallelogram are equal.
- The diagonals of a parallelogram bisect each other.
- The diagonal of a parallelogram divides it into two congruent triangles

A quadrilateral is said to a parallelogram if

Opposite sides are equal OR Opposite angles are equal OR Diagonal bisects each other OR A pair of opposite are parallel and equal

**Trapezium**

A quadrilateral which has one pair of opposite sides parallel is called a trapezium.

**Rhombus**

Rhombus is a parallelogram in which any pair of adjacent sides is equal.

Properties of a rhombus:

- All sides of a rhombus are equal
- The opposite angles of a rhombus are equal
- The diagonals of a rhombus bisect each other at right angles.

**Rectangle**

A parallelogram which has one of its angles a right angle is called a rectangle.

Properties of a rectangle are:

- The opposite sides of a rectangle are equal
- Each angle of a rectangle is a right-angle.
- The diagonals of a rectangle are equal.
- The diagonals of a rectangle bisect each other.

**Square**

A quadrilateral, all of whose sides are equal and all of whose angles are right angles.

Properties of square are:

- All the sides of a square are equal.
- Each of the angles measures 900.
- The diagonals of a square bisect each other at right angles.
- The diagonals of a square are equal.

All the quadrilaterals can be shown in Venn diagram like this

**Some important facts**

a) A square is always a parallelogram.

**Quadrilateral**

b) A square is always a rectangle.

c) A rhombus can be square.

e) A rectangle has four right angles.

**Mid-point Theorem for Triangles**

**Theorem-I**

The line segment joining the mid-points of the two sides of the triangle is parallel to the third side.

**Theorem-II**

A line drawn through mid-point of one side of a triangle and parallel to another side bisect the third side of the triangle.