Category : 9th Class

• A quadrilateral in which the measure of each angle is less than 180° is called a convex quadrilateral.

• A quadrilateral in which the measure of at least one of the angles is more than ${{180}^{o}}$s known as a concave quadrilateral.

• The sum of the angles of a quadrilateral is ${{360}^{o}}$ (or) 4 right angles.

• When the sides of a quadrilateral are produced, the sum of the four exterior angles so formed ${{360}^{o}}$ (i) Trapezium:

(a) A quadrilateral having exactly one pair of parallel sides is called a trapezium.

(b) A trapezium is said to be an isosceles trapezium if its non-parallel sides are equal.

ABCD is a trapezium in which AB || DC.

This trapezium is said to be an isosceles trapezium if AB || DC and AD = BC.

(ii) Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram.

ABCD is a parallelogram in which AB || DC and AD II BC.

Properties:

(a) In a parallelogram, any two opposite sides are equal.

(b) In a parallelogram, any two opposite angles are equal.

(c)In a parallelogram, the diagonals bisect each other.

(d) In a parallelogram, each diagonal divides it into two congruent triangles.

(e) In a parallelogram, any two adjacent angles have their sum equal to ${{180}^{o}}$ i.e. the adjacent angles are supplementary.

(iii) Rhombus: A quadrilateral having all sides equal is called a rhombus.

ABCD is a rhombus in which AB II DC, AD || BC and AB = BC = CD = DA.

Properties:

(a) The diagonals of a rhombus bisect each other at right angles.

(b) Each diagonal of a rhombus divides it into two congruent triangles.

(c) Opposite angles of a rhombus are equal and the sum of any two adjacent angles is${{180}^{o}}$

(d) The opposite sides of a rhombus are parallel.

(e) All the sides of a rhombus are equal.

(iv) Rectangle: A parallelogram whose angles are all right angles is called a rectangle.

ABCD is a rectangle in which,

AD || BC and AB || CD and

$\angle A=\angle B=\angle C=\angle D={{90}^{o}}$

• Properties:

(a) Opposite sides of a rectangle are equal and opposite angles of a rectangle are equal.

(b) The diagonals of a rectangle bisect each other.

(c) Each diagonal divides the rectangle into two congruent triangles.

(d) The diagonals of a rectangle are equal.

(v) Square: A parallelogram having all sides equal and each angle equal to a right angle is called a square.

ABCD is a square in which AB || DC, AD || BC,

AB = BC = CD = DA and ZA = ZB = ZC = ZD= 90°.

• Properties:

(a) All sides are equal.

(b) All angles are equal.

(c) The diagonals are equal and bisect each other at right angles.

(d) Each diagonal divides the square into two congruent right angled isosceles triangles.

(vi) Kite: A quadrilateral having two pairs of equal adjacent sides but unequal opposite sides is called a kite.

ABCD is a kite in which AB = AD and BC = CD.

• Conditions for a quadrilateral to become a parallelogram:

(a) Both pairs of opposite sides should be equal (or)

(b) One pair of opposite sides should be equal and parallel (or)

(c) Both pairs of opposite angles should be equal (or)

(d) Its diagonals should bisect each other.

• Conditions for a quadrilateral to become a rectangle:

(a) All its angles should be right angles (or)

(b) Its diagonals should be equal and bisect each other (or)

(c) Both pairs of opposite sides should be equal and one angle must be 90° (or)

(d) Both pairs of opposite sides and its diagonals should be equal.

Conditions for a quadrilateral to become a rhombus:

(a) All its sides should be equal (or)

(b) Its diagonals should bisect each other at right angles (or)

(c) Both pairs of its opposite sides should be equal and the diagonals should intersect at right angles.

• Conditions for a quadrilateral to become a square:

(a) All its sides should be equal and one angle must be 90° (or)

(b) All its sides and the diagonals should be equal (or)

(c) All its angles should be equal and the diagonals should intersect at right angles.

• Number of measurements required to construct different geometrical figures:

(i) To construct a quadrilateral, 5 independent measurements are needed.

(ii) To construct a trapezium, 4 independent measurements are required.

(iii) To construct a parallelogram, 3 independent measurements are required.

(iv) To construct a rhombus, 2 independent measurements are required.

(v) To construct a rectangle, 2 independent measurements are required.

(vi) To construct a square, 1 measurement is required.

• Midpoint theorem for a triangle:

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it.

• Converse: The line drawn through the midpoint of one side of a triangle parallel to another side of the triangle, bisects the third side of the triangle.
• The quadrilateral formed by joining the midpoints of the sides of a quadrilateral, in order, is a parallelogram.

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