Question

if three of the interior angles of a convex quadrilateral measure 109°, 139°, and 65°, find the measure of the fourth interior angle. 62° 42° 47° 56° none of these answers are correct. question 4 1 pts is it possible for the interior angles of a regular polygon to measure 23°? yes no

Explanation:

Step1: Recall the sum of interior angles of a quadrilateral

The sum of interior angles of a quadrilateral is \((4 - 2)\times180^{\circ}=360^{\circ}\)

Step2: Let the fourth angle be \(x\)

We know that \(109^{\circ}+139^{\circ}+65^{\circ}+x = 360^{\circ}\)

Step3: Solve for \(x\)

First, calculate \(109 + 139+65=313\). Then \(x=360 - 313=47^{\circ}\)

For the second question:

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

The measure of an interior angle \(\theta\) of a regular \(n\) - sided polygon is \(\theta=\frac{(n - 2)\times180^{\circ}}{n}=180^{\circ}-\frac{360^{\circ}}{n}\)
If \(\theta = 23^{\circ}\), then \(23^{\circ}=180^{\circ}-\frac{360^{\circ}}{n}\), so \(\frac{360^{\circ}}{n}=180^{\circ}- 23^{\circ}=157^{\circ}\), and \(n=\frac{360}{157}\approx2.29\). 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 part: \(47^{\circ}\)
For the second part: No