Understanding Quadrilaterals
Exercise 3.2
Question 1: Find x in the following figures:
Answer 1:
(a) Here, 125^{o}+m=180^{o} [Linear pair]
m=180^{o}125^{o} = 55^{o}
and 125^{o}+n =180^{o} [Linear pair]
n =180^{o}125^{o} = 55^{o}
Exterior angle x^{o} = Sum of opposite interior angles
x^{o} = 55^{o}+55^{o} =110^{o}
(b) Sum of angles of a pentagon = (n  2)x180^{o}
= (5  2)x180^{o}
= 3x180^{o} = 540^{o}
By linear pairs of angles,
Adding eq. (i), (ii), (iii), (iv) and (v),
Question 2:
Find the measure of each exterior angle of a regular polygon of:
(a) 9 sides
(b) 15 sides
Answer 2:
(i) Sum of angles of a regular polygon = (n2)x180^{o}
= (92)x180^{o} = 7x180^{o} =1260^{o}
Each interior angle= Sum of interior angles / Number of sides =1260^{o}/9 =140^{o}
Each exterior angle = 180^{o}140^{o} = 40^{o}
(ii) Sum of exterior angles of a regular polygon = 360^{o}
Each interior angle = Sum of interior angles/Number of sides = 360^{o}/15 =24^{o}
Question 3:
How many sides does a regular polygon have, if the measure of an exterior angle is 24^{o}?
Answer 3:
Let number of sides be n.
Hence, the regular polygon has 15 sides.
Question 4:
How many sides does a regular polygon have if each of its interior angles is 165?
Answer 4:
Let number of sides be n.
Exterior angle = 180^{o}165^{o=}15^{o}
Sum of exterior angles of a regular polygon = 360^{o}
Hence, the regular polygon has 24 sides.
Question 5:
(a) Is it possible to have a regular polygon with of each exterior angle as 22^{o}?
(b) Can it be an interior angle of a regular polygon? Why?
Answer 5:
(a) No. (Since 22 is not a divisor of 360^{o} )
(b) No, (Because each exterior angle is 180^{o}  22^{o} =158^{o}, which is not a divisor of 360^{o} )
Question 6:
(a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?
Answer 6:
(a) The equilateral triangle being a regular polygon of 3 sides has the least measure of an interior angle of 60^{o} .
Sum of all the angles of a triangle = 180^{o}
x+x+x= 180
3x=180^{0}
x=60^{o}
(b) By (a), we can observe that the greatest exterior angle is 180^{0}  60^{0} =120^{0}.
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