Question

a regular polygon is shown, with one of its angle measures labeled a. if $m\angle a = (5z + 65)\degree$, find the value of z.

Explanation:

Step1: Find the sum of interior angles

The formula for the sum of interior angles of a polygon is \((n - 2)\times180^{\circ}\), where \(n\) is the number of sides. For a non - agon (\(n=9\)), the sum is \((9 - 2)\times180^{\circ}=1260^{\circ}\).

Step2: Find the measure of one interior angle

Since it is a regular non - agon, each interior angle \(a=\frac{(n - 2)\times180^{\circ}}{n}\). Substituting \(n = 9\), we get \(a=\frac{1260^{\circ}}{9}=140^{\circ}\).

Step3: Solve for \(z\)

Given \(m\angle a=(5z + 65)^{\circ}\), and \(m\angle a = 140^{\circ}\). Then \(5z+65 = 140\). Subtract 65 from both sides: \(5z=140 - 65\), so \(5z=75\). Divide both sides by 5: \(z=\frac{75}{5}\).

Answer:

\(z = 15\)