Question

how many sides would a regular polygon have if each exterior angle measures ( 4^{circ} )?
______ sides
question 14
1 pts
if three of the interior angles of a convex quadrilateral measure ( 105^{circ}, 128^{circ} ), and ( 69^{circ} ), find the measure of the fourth interior angle.
______ degrees

Explanation:

Step1: Recall the formula for the sum of exterior angles of a polygon

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

Step2: Find the number of sides of the regular polygon

Let \(n\) be the number of sides. Since each exterior angle is \(4^{\circ}\), we use the formula \(n=\frac{360^{\circ}}{\text{measure of one exterior angle}}\). So \(n = \frac{360}{4}\).

Step3: Calculate \(n\)

\(n=90\).

Step4: Recall the formula for the sum of interior angles of a quadrilateral

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

Step5: Let the fourth interior angle be \(x\)

We know that \(105^{\circ}+128^{\circ}+69^{\circ}+x = 360^{\circ}\).

Step6: Solve for \(x\)

First, add the known angles: \(105 + 128+69=302\). Then \(x=360 - 302\).

Step7: Calculate \(x\)

\(x = 58^{\circ}\).

Answer:

For the first question: \(90\) sides.
For the second question: \(58\) degrees.