Question

The sum of the outer angles of a polygon is twice the sum of the inner angles.
(i) How many sides does it have?
(ii) What is the sum of outer angles and half the sum of inner angles?
(iii) If the sums are equal?
A) (i) 3 (ii) 4 (iii) 6
B) (i) 6 (ii) 4 (iii) 3
C) (i) 4 (ii) 3 (iii) 6
D) (i) 3 (ii) 6 (iii) 4

Answer 1

Hint:
It is given in the question that the sum of the outer angles of a polygon is twice the sum of the inner angles.
Then, we will find that
(i) How many sides does it have?
(ii) What is the sum of outer angles and half the sum of inner angles?
(iii) If the sums are equal?
Firstly, we have to find the Sum of interior angles of a polygon and Sum of exterior angles of a regular polygon.
Then, according to given questions we will find the values.

Complete step by step solution:
It is given in the question that the sum of the outer angles of a polygon is twice the sum of the inner angles.
Sum of interior angles of a polygon $ = \left( {n - 2} \right) \times {180^ \circ }$
 $\because $ Sum of exterior angles of a regular polygon $ = {360^ \circ }$
Since, it is given in the question that the sum of the outer angles of a polygon is twice the sum of the inner angles.
 $\therefore 2\left( {n - 2} \right) \times 180 \circ = {360^ \circ }$
 $\therefore \left( {n - 2} \right) \times {180^ \circ } = \dfrac{{{{360}^ \circ }}}{2}$
 $\therefore \left( {n - 2} \right){180^ \circ } = {180^ \circ }$
 $\therefore \left( {n - 2} \right) = 1$
 $\therefore n = 3$
 $ \Rightarrow $ The sum of outer angles is half the sum of inner angles
 $\therefore {360^ \circ } = \dfrac{1}{2}\left( {n - 2} \right) \times {180^ \circ }$
 $\therefore {360^ \circ } = \left( {n - 2} \right) \times {90^ \circ }$
 $\therefore \dfrac{{{{360}^ \circ }}}{{{{90}^ \circ }}} = \left( {n - 2} \right)$
 $\therefore \left( {n - 2} \right) = 4$
 $\therefore n = 6$
 $ \Rightarrow $ The sums are equal
 $\therefore \left( {n - 2} \right) \times {180^ \circ } = {360^ \circ }$
 $\therefore \left( {n - 2} \right) = \dfrac{{{{360}^ \circ }}}{{{{180}^ \circ }}}$
 $\therefore \left( {n - 2} \right) = 2$
 $\therefore n = 4$

Note:
Interior Angles: An angle formed between parallel lines by a third line that intersect them. An angle formed within a polygon by two adjacent sides.
Exterior Angles: An angle formed outside parallel lines by a third line that intersects them. An angle formed outside a polygon by one side and an extension of an adjacent side.