Question

1. what is the sum of the measures of the interior angles
2. what is the sum of the measures of the interior angles of a 25 - gon?
3. what is the measure of each interior angle of a regular hexagon?
4. what is the measure of each interior angle of a regular 16 - gon?
5. what is the sum of the measures of the exterior angles of a decagon?
6. what is the measure of each exterior angle of a regular 30 - gon?
7. an exterior angle of a regular polygon measures 22.5°. how many sides does it have?

Explanation:

Let's solve each sub - question one by one:

Sub - question 1: Sum of interior angles of a polygon (general formula)

The formula for the sum of the interior angles of a polygon with \(n\) sides is \((n - 2)\times180^{\circ}\). But since the number of sides \(n\) is not given here, we assume it's a general question about the formula. However, if we consider a polygon (let's assume a convex polygon, the formula holds for convex polygons), the sum \(S=(n - 2)\times180^{\circ}\). But maybe there is a typo or missing information. If we assume it's a general formula - based question, the sum of interior angles of a polygon is \((n - 2)\times180^{\circ}\), where \(n\) is the number of sides.

Sub - question 2: Sum of interior angles of a 25 - gon

Step 1: Recall the formula for the sum of interior angles of a polygon

The formula for the sum of interior angles of a polygon with \(n\) sides is \(S=(n - 2)\times180^{\circ}\).

Step 2: Substitute \(n = 25\) into the formula

For \(n = 25\), we have \(S=(25 - 2)\times180^{\circ}\)
\(S = 23\times180^{\circ}\)
\(S=4140^{\circ}\)

Sub - question 3: Measure of each interior angle of a regular hexagon

Step 1: Find the sum of interior angles of a hexagon

A hexagon has \(n = 6\) sides. Using the formula \(S=(n - 2)\times180^{\circ}\), we get \(S=(6 - 2)\times180^{\circ}=4\times180^{\circ} = 720^{\circ}\)

Step 2: Find the measure of each interior angle of a regular hexagon

In a regular polygon, all interior angles are equal. So each interior angle \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\). For \(n = 6\), \(\theta=\frac{720^{\circ}}{6}=120^{\circ}\)

Sub - question 4: Measure of each interior angle of a regular 16 - gon

Step 1: Find the sum of interior angles of a 16 - gon

Using the formula \(S=(n - 2)\times180^{\circ}\) with \(n = 16\), we have \(S=(16 - 2)\times180^{\circ}=14\times180^{\circ}=2520^{\circ}\)

Step 2: Find the measure of each interior angle

In a regular polygon, each interior angle \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\). For \(n = 16\), \(\theta=\frac{2520^{\circ}}{16}=157.5^{\circ}\)

Sub - question 5: Sum of exterior angles of a decagon

The sum of the exterior angles of any convex polygon (and regular polygons are convex) is always \(360^{\circ}\), regardless of the number of sides. So for a decagon (a 10 - sided polygon), the sum of exterior angles is \(360^{\circ}\)

Sub - question 6: Measure of each exterior angle of a regular 30 - gon

Step 1: Recall the sum of exterior angles of a polygon

The sum of exterior angles of any convex polygon is \(360^{\circ}\).

Step 2: Find the measure of each exterior angle of a regular 30 - gon

In a regular polygon, all exterior angles are equal. So each exterior angle \(\alpha=\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. For \(n = 30\), \(\alpha=\frac{360^{\circ}}{30}=12^{\circ}\)

Sub - question 7: Number of sides of a regular polygon with each exterior angle \(22.5^{\circ}\)

Step 1: Recall the formula for the measure of each exterior angle of a regular polygon

The measure of each exterior angle \(\alpha=\frac{360^{\circ}}{n}\), where \(n\) is the number of sides.

Step 2: Solve for \(n\)

We know that \(\alpha = 22.5^{\circ}\) and \(\alpha=\frac{360^{\circ}}{n}\). So we can re - arrange the formula to \(n=\frac{360^{\circ}}{\alpha}\)
Substitute \(\alpha = 22.5^{\circ}\) into the formula: \(n=\frac{360^{\circ}}{22.5^{\circ}} = 16\)

Answer:

s:

1. The sum of interior angles of a polygon is \((n - 2)\times180^{\circ}\) (where \(n\) is the number of sides).
2. The sum of interior angles of a 25 - gon is \(4140^{\circ}\)
3. The measure of each interior angle of a regular hexagon is \(120^{\circ}\)
4. The measure of each interior angle of a regular 16 - gon is \(157.5^{\circ}\)
5. The sum of exterior angles of a decagon is \(360^{\circ}\)
6. The measure of each exterior angle of a regular 30 - gon is \(12^{\circ}\)
7. The regular polygon with each exterior angle \(22.5^{\circ}\) has 16 sides.