Question

1. answer the questions about the polygons. one of them is a trick question; for that problem, explain why it is a trick question.

a) if a regular polygon has 30 sides, what is the measure of each exterior angle?
b) the degree measure of each exterior angle of a regular octagon is represented by the expression 7x - 4. solve for x.
c) what is the sum total of the interior angles of a nonagon?

Explanation:

Step1: Use the formula for exterior angles of a regular polygon

The sum of exterior angles of any polygon is \(360^{\circ}\). For a regular polygon with \(n\) sides, the measure of each exterior angle \(E=\frac{360^{\circ}}{n}\).
Given \(n = 30\), then \(E=\frac{360^{\circ}}{30}\)

Step2: Calculate the value

\(E = 12^{\circ}\)

Step3: For part b)

A regular octagon has \(n=8\) sides. Using the formula \(E=\frac{360^{\circ}}{n}\), we substitute \(n = 8\). So \(E=\frac{360^{\circ}}{8}=45^{\circ}\). Then we set \(7x - 4=45\)

Step4: Solve the equation for \(x\)

Add \(4\) to both sides: \(7x=45 + 4=49\). Divide both sides by \(7\): \(x=\frac{49}{7}=7\)

Step5: For part c)

Use the formula for the sum of interior angles of a polygon \(S=(n - 2)\times180^{\circ}\). For a non - agon (\(n = 9\)), \(S=(9 - 2)\times180^{\circ}\)

Step6: Calculate the sum

\(S=7\times180^{\circ}=1260^{\circ}\)

Answer:

a) \(12^{\circ}\)
b) \(x = 7\)
c) \(1260^{\circ}\)