Question

37 for the diagram, determine the number of sides use the polygon angle sum theorem to find the sum of the interior angles; (show all work on paper) n = sum of interior angles = degrees.

Explanation:

Step1: Recall Polygon Angle Sum Theorem

The Polygon Angle Sum Theorem states that the sum of the interior angles of a polygon with \( n \) sides is given by the formula \( S=(n - 2)\times180^{\circ} \), where \( n \) is the number of sides of the polygon. But first, we need to know the number of sides \( n \). Wait, maybe the diagram (not shown here) has a polygon, for example, if it's a pentagon, \( n = 5 \); if it's a hexagon, \( n=6 \), etc. Let's assume we can determine \( n \) from the diagram. Let's suppose the diagram is of a pentagon (just an example, but since the diagram is not visible, maybe a common case). Wait, maybe the original problem (if it's a standard problem) has a polygon, say, let's take a common one. Wait, maybe the user missed the diagram, but let's proceed with the formula. Let's assume \( n \) is known from the diagram. Let's say the diagram is a pentagon, so \( n = 5 \).

Step2: Apply the formula

If \( n = 5 \) (pentagon), then using the formula \( S=(n - 2)\times180^{\circ} \), we substitute \( n = 5 \):
\( S=(5 - 2)\times180^{\circ}=3\times180^{\circ} = 540^{\circ} \)

But wait, the problem says "determine the number of sides" first. So first, we need to find \( n \) from the diagram. Since the diagram is not shown, but maybe in the original problem, the diagram is, for example, a hexagon (6 sides) or pentagon (5 sides), etc. Let's assume that from the diagram, the number of sides \( n \) is, say, 5 (pentagon). Then:

For \( n = 5 \):
Sum of interior angles \( S=(5 - 2)\times180^{\circ}=540^{\circ} \)

Or if \( n = 6 \) (hexagon):
\( S=(6 - 2)\times180^{\circ}=4\times180^{\circ}=720^{\circ} \)

But since the diagram is not provided, maybe there was a typo or the diagram is a regular polygon. Wait, the problem says "determine the number of sides" first. So first, identify \( n \) from the diagram (count the sides). Then apply the formula.

Let's suppose the diagram is a pentagon (5 sides). Then:

Step1: Determine \( n \)

Count the sides of the polygon from the diagram. Let's say it's a pentagon, so \( n = 5 \).

Step2: Apply Polygon Angle Sum Theorem

Use \( S=(n - 2)\times180^{\circ} \). Substitute \( n = 5 \):
\( S=(5 - 2)\times180^{\circ}=3\times180 = 540 \)

So if \( n = 5 \), then \( n = 5 \) and sum is \( 540 \) degrees.

(Note: Since the diagram is not shown, the actual \( n \) depends on the diagram. But the process is: 1. Find \( n \) by counting sides. 2. Use \( S=(n - 2)\times180 \) to find the sum.)

Answer:

Assuming the diagram is a pentagon (5 sides):
\( n = \boldsymbol{5} \)
Sum of Interior Angles = \(\boldsymbol{540}\) degrees.

(If the diagram has a different number of sides, adjust \( n \) accordingly. For example, if it's a hexagon, \( n = 6 \) and sum is \( 720 \) degrees.)