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Types of Quadrilaterals and other Polygons

Quadrilaterals

A closed plane figure bounded by four line segments is called quadrilateral.

In the adjoining diagram, ABCD is a quadrilateral and it has four sides – AB, BC, CD and DA. It has four (interior) angles – angle A, angle B, angle C and angle D and four vertices – A, B, C, and D. It has two diagonals – AC and BD.

Quadrilaterals and other polygons

Sum of (interior) angles of a quadrilateral is 3600



In the adjoining figure, ABCD is any quadrilateral and diagonal AC divides it into two triangles. You know that the sum of angles of a triangle is 1800,

from ? ABC, angle 1 + angle B + angle 2 = 1800 …..(i)

from ? ACD, angle 4 + angle D + angle 3 = 1800 ……(ii)

on adding (i) and (ii), we get

angle 1 + angle 4 + angle B + angle D + angle 2 + angle 3 = 3600

? angle A + angle B + angle D + angle C = 3600 (from figure)

Hence the sum of (interior) angles of a quadrilateral is 3600

Quadrilaterals and other polygons


Parallelogram

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

In the adjoining quadrilateral, AB ?? DC and AD ?? BC, so ABCD is a parallelogram.


Quadrilaterals and other polygons

Theorem 1

  1. The opposite sides of a parallelogram are equal.
  2. The opposite angles of a parallelogram are equal.
  3. Each diagonal bisects the parallelogram.

Quadrilaterals and other polygons

Theorem 2

The diagonals of a parallelogram bisect each other


Quadrilaterals and other polygons

Theorem 3

If a pair of opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.


Quadrilaterals and other polygons


Regular Polygon

A polygon is called regular polygon if all its sides have equal length and all its angles have equal size.

Thus, in a regular polygon:

  • All sides are equal in length
  • All interior angles are equal in size
  • All exterior angles are equal in size

All regular polygon are convex

Angle Property of a Polygon

Sum of interior angles of a polygon

In the adjoining figure, ABCDE is a pentagon. It has 5 sides and 5 (interior) angles. Take any point O inside the pentagon and join it with vertices. We notice that 5 triangles are formed.

As the sum of angles of a triangle is 2 right angles, therefore, the sum of all the angles of the 5 triangles = (2 × 5) right angles.

Also the sum of angles at the point O = 4 right angles. It follows that the sum of all the (interior) angles of the pentagon ABCDE = (2 × 5 – 4) right angles. In fact, this is true about every polygon of n sides. So, we have an important.


Quadrilaterals and other polygons

Example

Example 1

(i)Find the sum of interior angles of nonagon

(ii) Find the measure of each interior angles of a regular 16-gon

Solution

(i) A nonagon has 9 sides

Sum of its interior angles = (2 × 9 – 4) right angles

= 14 × 900 = 12600

(ii) Each interior angle of a regular 16-sided polygon

= 3600/16 = 450/2 = 22.50 = 220 30’

... Each interior angle of regular 16-gon = 1800 – 220 30’ = 1570 30’

Example 2

A heptagon has 4 equal angles each of 1320 and three equal angles. Find the size of equal angles.

Solution.

A heptagon has 7 sides

Sum of its interior angles = (2 × 9 – 4) right angles

= 10 × 900 = 9000

Let the size of each equal angle be x, so we have

= 4 × 1320 + 3x = 9000

? 3x = 9000 – 5280 = 3720 ? x = 1240

Hence the size of each equal angle = 1240

Example3

The sum of interior angles of a polygon is 27000. How many sides this polygon has?

Solution.

Let the polygon have n sides, then the sum of its interior angles

= (2n – 4) right angles = (2n – 4) × 900

By the question, (2n – 4) × 900 = 27000

? 2n – 4 = 30 ? 2n = 34 ? n = 17

Hence the polygon has 17 sides


Solve the Question

1. Each interior angle of a regular polygon is double of its exterior angle. Find the number of sides in

the polygon?

2. Each interior angle of a regular polygon is 1500. Find the interior angle of a regular polygon which

has double the number of sides as the given polygon?

3. In the adjoining figure, ABCDE is a regular pentagon. Find

(i) angle ABC (ii) angle CAB (iii) angle ACD


Quadrilaterals and other polygons