Question

d) if all of the angles of a decagon are equal in measure, how much is each exterior angle?
e) the sum total of all of the exterior angles of a polygon is 360°. how many sides does it have?
f) if each interior angle of a regular polygon measures 174°, how many sides does the polygon have?

Explanation:

Step1: Recall formula for exterior angles of regular polygon

For a regular polygon, the sum of exterior angles is \(360^{\circ}\). If all exterior angles are equal, and let \(n\) be the number of sides (also the number of exterior angles). The measure of each exterior angle \(x=\frac{360^{\circ}}{n}\). For a decagon (\(n = 10\)), \(x=\frac{360^{\circ}}{10}\)

Step2: Calculate the measure of each exterior angle

\(\frac{360^{\circ}}{10}=36^{\circ}\)

Step3: Recall formula for sum of exterior angles of any polygon

The sum of exterior angles of any polygon (regardless of the number of sides) is \(360^{\circ}\)

Step4: Use the relationship between interior and exterior angles

Let the measure of an interior angle be \(I\) and an exterior angle be \(E\). We know that \(I + E=180^{\circ}\). If \(I = 174^{\circ}\), then \(E=180^{\circ}-174^{\circ}=6^{\circ}\)
Since the sum of exterior angles \(S = 360^{\circ}\) and \(S=n\times E\) (where \(n\) is the number of sides), then \(n=\frac{360^{\circ}}{E}\)
Substitute \(E = 6^{\circ}\) into the formula: \(n=\frac{360^{\circ}}{6^{\circ}}\)

Step5: Calculate the number of sides

\(\frac{360^{\circ}}{6^{\circ}}=60\)

Answer:

d) \(36^{\circ}\)
e) \(360^{\circ}\)
f) \(60\)