Question

1. * required each angle of a regular n - gon measures 172.8°. how many sides does the n - gon have? corollary to theorem 6 - 1 the measure of an interior angle of a regular n - gon is ( 180 cdot \frac{(n - 2)}{n} ). a pentagon (5 sides) a 50 - gon a 180 - gon a decagon

Explanation:

Step1: Recall the formula for the measure of an interior angle of a regular n - gon.

The formula for the measure of each interior angle of a regular \(n\) - gon is \(I=\frac{(n - 2)\times180^{\circ}}{n}\), where \(I\) is the measure of each interior angle and \(n\) is the number of sides. We know that \(I = 172.8^{\circ}\), so we set up the equation \(\frac{(n - 2)\times180}{n}=172.8\).

Step2: Solve the equation for \(n\).

First, multiply both sides of the equation by \(n\) to get \((n - 2)\times180=172.8n\).
Expand the left - hand side: \(180n-360 = 172.8n\).
Subtract \(172.8n\) from both sides: \(180n-172.8n-360=0\).
Simplify the left - hand side: \(7.2n-360 = 0\).
Add 360 to both sides: \(7.2n=360\).
Divide both sides by 7.2: \(n=\frac{360}{7.2}=50\).

Answer:

A 50 - gon