We are asked to use two different methods of finding the interior angles of a regular polygon. Let's start!
Method 1
If we know the number of sides in a polygon n, the sum of its angles can be calculated using the following formula.
180^(∘)(n-2)
Furthermore, if a polygon is regular all of its interior angles are congruent. This means if we divide the formula by the number of sides n the quotient should equal the measure of an interior angle in the polygon.
180^(∘)(n-2)/n
Let's substitute n=30 in this formula and simplify.
The interior angle of the regular 30-gon is 168^(∘).
Method 2
The exterior angles of an n-gon always sum to 360^(∘). Since the exterior angles of a regular polygon are congruent, we can determine the measure of the exterior angles by dividing 360^(∘) by 30.
360^(∘)/30=12^(∘)
The exterior angles of the polygon are 12^(∘). Also, the exterior and corresponding interior angle in a polygon form a linear pair. This means we can write the following equation.
12^(∘)+m∠ θ = 180^(∘) ⇔ m∠ θ = 168^(∘)
We arrived at the same result as with the first method.
