Question

how many sides would a regular polygon have if each interior angle measures 172°?
______ sides

question 10
1 pts
calculate the measure for each exterior angle of a regular decagon.**round your answer to two decimal places if necessary.
______ degrees

Explanation:

First problem (number of sides of a regular polygon with interior angle \(172^{\circ}\))

Step1: Find the measure of the exterior angle

The sum of an interior angle and its corresponding exterior angle of a polygon is \(180^{\circ}\).
Let \(x\) be the measure of the exterior angle. Then \(x = 180 - 172=8^{\circ}\)

Step2: Use the formula for the measure of an exterior angle of a regular polygon

The formula for the measure of an exterior angle \(\theta\) of a regular polygon with \(n\) sides is \(\theta=\frac{360^{\circ}}{n}\)
Since \(\theta = 8^{\circ}\), we have the equation \(8=\frac{360}{n}\)
Cross - multiply: \(8n=360\)
Solve for \(n\): \(n=\frac{360}{8}=45\)

Second problem (measure of an exterior angle of a regular decagon)

Step1: Recall the formula for the measure of an exterior angle of a regular polygon

The formula for the measure of an exterior angle \(\theta\) of a regular polygon with \(n\) sides is \(\theta=\frac{360^{\circ}}{n}\)
A decagon has \(n = 10\) sides.
Substitute \(n = 10\) into the formula: \(\theta=\frac{360}{10}=36^{\circ}\)

Answer:

For the first problem: \(45\)
For the second problem: \(36\)