PROPERTIES OF QUADRILATERAL GEOMETRY

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A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices and four angles.

Sum of interior angles of a quadrilateral is 360.

There are six types of quadrilaterals. They are

Parallelogram

Definition of parallelogram :

A parallelogram is a quadrilateral which has opposite sides parallel.

Properties of parallelogram :

Opposite sides are equal in length
Opposite angles are equal in size.
Diagonals bisect each other.

Rectangle

Definition of rectangle :

A rectangle is a parallelogram with four equal angles of 90 degree.

Properties of rectangle :

Opposite sides are equal and parallel.
Diagonals bisect each other.
Diagonals are equal in length

Square

What is square ?

A square is a rhombus with four equal angles of 90.

Properties :

Opposite sides are parallel.
All sides are equal in length.
Diagonals bisect each other at right angles.
Diagonals bisect the angles at each vertex.
Diagonals are equal in length.

Trapezium

What is trapezium ?

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

Properties :

A trapezium has two parallel sides and two non parallel sides.
The diagonal of regular trapezium bisect each other.
The length of the mid segment is equal to half the sum of parallel bases.
Two pairs of adjacent angles of a trapezium formed between the parallel sides and one of the non parallel side, add up to 180 degrees.

Kite

To find diagonal, we have the following ways.

(i) From the given area and one diagonal, find the other diagonal.

(ii) Using Pythagorean theorem, find length of diagonal.

The diagonals of a kite are perpendicular to each other. The longer diagonal of the kite bisects the shorter diagonal.

Area of kite ?

A kite is a quadrilateral which has two pairs of adjacent sides equal in length.

To find area of kite we need diagonals.

Area of kite = (1/2) x diagonal 1 x diagonal 2

Properties of kite :

If a quadrilateral is a kite, then its diagonals are perpendicular. AC is perpendicular to BD.
If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If quadrilateral ABCD is a kite and BC ≅ BA , then ∠A ≅ ∠C and ∠B ≇ ∠D.

Problem 1 :

One angle of a quadrilateral is 150° and other three angles are equal. What is the measure of each of these equal angles ?

(a) 75° (b) 85° (c) 95° (d) 70°

Solution :

Let x be the equal angles.

150 + x + x + x = 360

3x + 150 = 360

3x = 360 - 150

3x = 210

x = 70

So, the equal angle measure will be 70 degree.

Problem 2 :

Two adjacent sides AB and BC of a parallelogram ABCD are in the ratio 5 : 3. If the perimeter is 200 cm, what is the length of AB and BC ?

(a) 25 cm & 50 cm (b) 40 cm & 37.5 cm

Solution :

Perimeter = 200 cm

Sides are in the ratio 5 : 3

Side lengths of the parallelogram will be 5x and 3x.

2(5x + 3x) = 200

2(8x) = 200

8x = 100

x = 100/8

= 12.5

5x = 5(12.5) ==> 62.5

3x = 3(12.5) ==> 37.5

So, the side lengths are 62.5 cm and 37.5 cm.

Problem 3 :

Fill the blanks for the following statements :

A parallelogram having all sides equal is called a ................ .

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

If the consecutive sides of a parallelogram are equal, then it is necessarily a ................. .

Solution :

A parallelogram having all sides equal is called a rhombus.

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

If the consecutive sides of a parallelogram are equal, then it is necessarily a rhombus

PROPERTIES OF QUADRILATERAL GEOMETRY

Parallelogram

Rectangle

Square

Trapezium

Kite

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