Question

proving the polygon exterior angle sum theorem
given: a convex n - gon
prove: the sum of the measures of the exterior angles is 360°.
complete the missing parts of the paragraph proof.
for a convex n - gon, interior and exterior angles form angles, whose measures sum to 180°.
because we have an n - gon, the sum of the measures of these linear pairs, one exterior angle at each vertex, is .
the sum of the measures of the interior angles is by the polygon interior angle sum theorem.
to find the measure of the exterior angles, we can subtract the sum of the measures of the interior angles from the sum of the measures of the linear pairs. therefore, we have , which is equal to 360°.

Explanation:

Step1: Identify the type of angle pair

Interior and exterior angles form linear - pair angles. A linear pair of angles is adjacent angles that are supplementary (sum to \(180^{\circ}\)).

Step2: Calculate the sum of linear - pair angles for an \(n\) - gon

Since each of the \(n\) vertices has a linear pair of angles (interior + exterior) and each linear pair sums to \(180^{\circ}\), the sum of the measures of these linear pairs is \(n\times180^{\circ}\).

Step3: Recall the formula for the sum of interior angles

By the polygon interior - angle sum theorem, the sum of the measures of the interior angles of an \(n\) - gon is \((n - 2)\times180^{\circ}=180n-360\).

Step4: Derive the sum of exterior angles

Let \(S_{ext}\) be the sum of exterior angles. We know that \(S_{ext}+S_{int}=n\times180^{\circ}\) (where \(S_{int}\) is the sum of interior angles). Substituting \(S_{int}=(n - 2)\times180^{\circ}\) into the equation:

$$
LATEXBLOCK0
$$

The blanks are filled as follows:

1. linear - pair
2. \(n\times180^{\circ}\)
3. \((n - 2)\times180^{\circ}\)
4. \(n\times180^{\circ}-(n - 2)\times180^{\circ}\)

Answer:

1. linear - pair
2. \(n\times180^{\circ}\)
3. \((n - 2)\times180^{\circ}\)
4. \(n\times180^{\circ}-(n - 2)\times180^{\circ}\)