b The interior angle and corresponding exterior angle of a polygon form a linear pair. Therefore, if the exterior angle of a regular polygon is 42^(∘), we can find the interior angle, θ, by equating the measure of these angles with 180^(∘) and solving for m∠θ.
m∠ θ + 42^(∘)= 180^(∘) ⇔ m∠ θ = 138^(∘)The interior angle is 138^(∘). Like in Part A, we can find the sum of a polygons interior angles by using the formula 180^(∘)(n-2). In a regular polygon, all angles have the same measure. Therefore, if one interior angle is 138^(∘) and we have n sides, we can write the following equation.
180^(∘)(n-2)=138^(∘) n
Let's solve for n in this equation.
Since we did not get an integer for n, this type of polygon is not possible.
c A pentagon is a quadrilateral with 5 sides. Assuming that the pentagon is regular, we can find the measure of each angle by substituting n=5 in the formula 180^(∘)(n-2)n and simplifying. However, it is not stated that the polygon is regular, so we cannot find the interior angles.
d Using the same formula as in Part C we can find the measure of each interior angle in the regular decagon, which is a polygon with 10 congruent sides.
