When you walk around a polygon, at each vertex you need to turn either right (clockwise) or left (counterclockwise). A turn to the left is measured by a positive number of degrees and a turn to the right by a negative number of degrees. Find the sum of the measures of the turn angles of the polygons you drew. Assume you start at a vertex facing in the direction of a side, walk around the polygon, and end up at the same vertex facing in the same direction as when you started

Question

When you walk around a polygon, at each vertex you need to turn either right (clockwise) or left (counterclockwise). A turn to the left is measured by a positive number of degrees and a turn to the right by a negative number of degrees. Find the sum of the measures of the turn angles of the polygons you drew. Assume you start at a vertex facing in the direction of a side, walk around the polygon, and end up at the same vertex facing in the same direction as when you started

Answer 1

I don't know what polygons you drew, but consider this: no matter what path you took, if at any point you are turned to the left of where you started, you must have turned through 90 degrees (plus some integer number of full turns); if at any point you are turned backward compared to where you started, you must have turned through 180 degrees (plus some number of full turns).

And finally, if you are facing the direction you started in, you must have turned through zero degrees (plus some integer number of full turns).

Answer 2

I think the purpose of your experiment was to show that for any polygon, the sum of the exterior angles will always be 360º

(the simplest example would be to totally walk around a normal city block. You will have made 4 90º turns.)

Answer 3

To find the sum of the measures of the turn angles of a polygon, you can use the fact that the sum of the interior angles of a polygon with n sides is given by the formula (n - 2) * 180 degrees.

Let's say you have a polygon with n sides. As you walk around the polygon, you start at a vertex facing in the direction of a side and end up at the same vertex facing the same direction. This means that you have walked around the polygon in a complete loop, covering a total of n sides.

Since each turn at a vertex corresponds to an interior angle, we can say that the sum of the turn angles is equal to the sum of the interior angles of the polygon. Using the formula mentioned earlier, we have:

Sum of turn angles = Sum of interior angles = (n - 2) * 180 degrees

Therefore, to find the sum of the measures of the turn angles of the polygon you drew, substitute the number of sides of your polygon into the formula (n - 2) * 180 degrees.