Question 2: Do you feel it is true that every 5-sided polygon subdivides into three triangles? If so, what does this say is the sum of the measures of the interior angles of a five-sided polygon?

Question 3: Do you feel it is true that every 13-sided polygon subdivides into eleven triangles? If so, what does this say is the sum of the measures of the interior angles of a 13-sided polygon?

What general formula might you suggest for the sum of measures of the interior angles of a polygon with \(N\) sides?

Question 4: The beginning assumptions in the theory of geometry show that for any straight line, such as the one shown in blue, with two angles constructed on it as shown have measures summing to \(180^{\circ}\). (This feels intuitively correct!)

a)  Use this next diagram to show that pencil-pushing is correct in suggesting that vertical angles must have the same measure. (Explain why the green and the purple angles have the same measure.)

b) Explain why the sum of the measures of the green and the purple angles in this diagram equals e measure of the blue angle.

c) A hexagon subdivides into four triangles and so the measures of its six interior angles sum to \(4 \times 180^{\circ}\). In this diagram where these measures are denoted \(a\), \(b\), \(c\), \(d\), \(e\), and  \(f\) and we have \(a+b+c+d+e+f=4 \times 180^{\circ}\).

Use this to show that pencil pushing was correct to say that the six exterior angles (shown as dots) have measures that sum to \(360^{\circ}\).

d)  CHALLENGE: Show that pencil-pushing was correct to say that the measures of the angles shown in a lopsided star (represented as letters) add to half a turn, that is, \(a+b+c+d+e=180^{\circ}\).

(Hint: Why is the measure of the angle with the dot equal to \(b+e\)?)