1. Find x in the following figures.
a)
Solution:
125° + m = 180° [Linear pair]
m = 180° – 125° = 55°
125° + n = 180° [Linear pair]
n = 180° – 125° = 55°
we know that exterior angle of a triangle is equal to the sum of 2 opposite interior 2 angles
x = m + n
x = 55° + 55° = 110°
b)
Two interior angles are right angles = 90°
70° + m = 180° [Linear pair]
m = 180° – 70° = 110°
60° + n = 180° [Linear pair]
n = 180° – 60° = 120°
The given figure is having five sides and is a pentagon.
Thus, sum of the angles of pentagon = 540°
90° + 90° + 110° + 120° + y = 540°
410° + y = 540° ⇒ y = 540° – 410° = 130°
x + y = 180° (Linear pair)
x + 130° = 180°
x = 180° – 130° = 50°
2. Find the measure of each exterior angle of a regular polygon of
(i) 9 sides (ii) 15 sides
Solution:
We know that sum of angles a regular polygon having side n = (n-2)×180°
(i)here n = 9
Sum of angles a regular polygon having side 9 = (9-2)×180°= 7×180° = 1260°
Each interior angle=1260/9 = 140°
Each exterior angle = 180° – 140° = 40°
(ii) here n =15
Sum of angles a regular polygon having side 15 = (15-2)×180°
13×180° = 2340°
Each interior angle = 2340/15 = 156°
Each exterior angle = 180° – 156° = 24°
3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
Solution:
Each exterior angle = sum of exterior angles/Number of angles
24°= 360°/ Number of sides
Number of sides = 360°/24 = 15
Hence, the regular polygon has 15 sides.
4. How many sides does a regular polygon have if each of its interior angles is 165°?
Solution:
Given, Interior angle = 165°
So,
Exterior angle = 180° – 165° = 15°
Number of sides = sum of exterior angles/ exterior angles
Number of sides = 360/15 = 24
Hence, the regular polygon has 24 sides.
5.
a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
Solution:
Given, Exterior angle = 22°
Now,
Number of sides = sum of exterior angles/ exterior angle
Number of sides = 360/22 = 16.36
No, we can’t have a regular polygon with each exterior angle as 22° because it is not divisor of 360°.
b) Can it be an interior angle of a regular polygon? Why?
Solution:
Interior angle = 22°
Exterior angle = 180° – 22°= 158°
No, we can’t have a regular polygon with each exterior angle as 158° because it is not divisor of 360°.
6.
a) What is the minimum interior angle possible for a regular polygon? Why?
Solution:
As we know that Equilateral triangle is regular polygon which can be constructed with 3 sides at least. Since, sum of interior angles of a triangle = 180°
Hence,
Sum of all interior angles of a regular polygon of side n = (n-2) x 180°
The measure of each interior angle = ((n-2) x 180°)/n
For minimum possible interior angle:
((n-2) x 180°)/n > 0
n>0
Hence, the minimum measure of the angle of an equilateral triangle (n=3)=60°.
b) What is the maximum exterior angle possible for a regular polygon?
Solution:
As we know that Equilateral triangle is regular polygon with 3 sides.
From the above (part a), it conclude that the maximum exterior angle of a regular polygon = 180° – 60° = 120°
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