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5.(a) Is it possible to have a regular polygon with measure of each exterior angle a is \(22^{\circ}\)?
(b) Can it be an interior angle of a regular polygon? Why?

Answer :

(a) We have, Number of sides of regular polygon=\(\frac{360^{\circ} }{Exterior \;angle}\)

As we know number of sides by the formula should be an integer so, it is not possible that a regular polygon have its exterior angle of \(22^{\circ}\) because \(22^{\circ}\) is not divisible by \(360^{\circ}\) hence, we can't get a whole number for calculating the number of side.

(b) If interior angle =\(22^{\circ}\)

And we have: measure of each interior angle=\(\frac{(n-2)\times180^o}n\)

\(\Rightarrow 180n -2 \times 180=22n\)

\(\Rightarrow 180n -22n=360\)

\(\Rightarrow 158n=360\)

\(\Rightarrow n=\frac{360}{158}\)

But 158 does not divide 360 exactly.So, the polygon is not possible.

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