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.
Sum of (interior) angles of a quadrilateral is 360^{0}

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 180^{0},
from ? ABC, angle 1 + angle B + angle 2 = 180^{0} …..(i)
from ? ACD, angle 4 + angle D + angle 3 = 180^{0} ……(ii)
on adding (i) and (ii), we get
angle 1 + angle 4 + angle B + angle D + angle 2 + angle 3 = 360^{0}
? angle A + angle B + angle D + angle C = 360^{0} (from figure)
Hence the sum of (interior) angles of a quadrilateral is 360^{0}
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.
Theorem 1
 The opposite sides of a parallelogram are equal.
 The opposite angles of a parallelogram are equal.
 Each diagonal bisects the parallelogram.
Theorem 2
The diagonals of a parallelogram bisect each other
Theorem 3
If a pair of opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.
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
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.
Example
Example 1
(i)Find the sum of interior angles of nonagon
(ii) Find the measure of each interior angles of a regular 16gon
Solution
(i) A nonagon has 9 sides
Sum of its interior angles = (2 × 9 – 4) right angles
= 14 × 90^{0} = 1260^{0}
(ii) Each interior angle of a regular 16sided polygon
= 360^{0}/16 = 45^{0}/2 = 22.5^{0} = 22^{0} 30’
.^{.}. Each interior angle of regular 16gon = 180^{0} – 22^{0} 30’ = 157^{0} 30’
Example 2
A heptagon has 4 equal angles each of 132^{0} 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 × 90^{0} = 900^{0}
Let the size of each equal angle be x, so we have
= 4 × 132^{0} + 3x = 900^{0}
? 3x = 900^{0} – 528^{0} = 372^{0} ? x = 124^{0}
Hence the size of each equal angle = 124^{0}
Example3
The sum of interior angles of a polygon is 2700^{0}. 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) × 90^{0}
By the question, (2n – 4) × 90^{0} = 2700^{0}
? 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 150^{0}. 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