NCERT Class 9 Mathematics Quadrilaterals CBSE Board Sample Problems

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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

A Quadrilateral

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A quadrilateral ¡s 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

The opposite sides of a parallelogram are equal

The Opposite Sides of a Parallelogram Are Equal

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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.

Called a trapezium

Called a Trapezium

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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.

Properties of a rhombus

Properties of a Rhombus

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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.

Called a rectangle

Called a Rectangle

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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.

The diagonals of a square are equal

The Diagonals of a Square Are Equal

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All the quadrilaterals can be shown in Venn diagram like this

Some important facts

a) A square is always a parallelogram.

Some important facts

Some Important Facts

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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.

Mid-point Theorem for Triangles

Mid-Point Theorem for Triangles

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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.