Question

if, for a convex octagon, four exterior angles each measure 22° and three exterior angles each measure 33°, find the measure of the eighth exterior angle. 181° 172° 173° 167° none of these answers are correct. question 2 1 pts it is possible for the interior angles of a regular polygon to measure 91°. true false

Explanation:

Step1: Recall the sum of exterior angles of a polygon

The sum of exterior angles of any convex polygon is \(360^{\circ}\).

Step2: Calculate the sum of the known exterior angles

Four exterior angles each of \(22^{\circ}\) contribute \(4\times22^{\circ}=88^{\circ}\). Three exterior angles each of \(33^{\circ}\) contribute \(3\times33^{\circ} = 99^{\circ}\). The sum of these seven angles is \(88^{\circ}+99^{\circ}=187^{\circ}\).

Step3: Find the measure of the eighth exterior angle

Let the measure of the eighth exterior angle be \(x\). Using the sum of exterior angles formula \(x+187^{\circ}=360^{\circ}\). Solving for \(x\), we get \(x = 360^{\circ}-187^{\circ}=173^{\circ}\).

For the second question:

Step1: Recall the formula for the interior angle of a regular polygon

The formula for the interior angle \(\theta\) of a regular polygon with \(n\) sides is \(\theta=\frac{(n - 2)\times180^{\circ}}{n}=180^{\circ}-\frac{360^{\circ}}{n}\).

Step2: Assume \(\theta = 91^{\circ}\) and solve for \(n\)

If \(91^{\circ}=180^{\circ}-\frac{360^{\circ}}{n}\), then \(\frac{360^{\circ}}{n}=180^{\circ}-91^{\circ}=89^{\circ}\), and \(n=\frac{360^{\circ}}{89^{\circ}}\approx4.04\). Since \(n\) (the number of sides of a polygon) must be a positive integer greater than or equal to 3, it is not possible.

Answer:

For the first question: \(173^{\circ}\) (the third option).
For the second question: False.