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In Part C, we explore the interior and exterior angles of a figure as well as the relationship between the two. This activity will provide you with a different method of exploring angle measurement.

**Problem C1**

On a piece of paper draw the following shapes. Pay attention to the amount of turn you do with your pencil as you draw any two adjacent sides (the amount of turn will be equal to the exterior angle of the polygon at that vertex).

- Square
- Rectangle
- Parallelogram that is not a rectangle
- Equilateral triangle
- Regular pentagon
- Regular hexagon
- Star polygon (five- or six-pointed star)
- n-gon (you decide the number for n)

**Problem C2 **

- What is the relationship between the amount of a turn and the resulting interior angle?
- Why is this relationship important to understand?

**Problem C3**

Examine the polygons you drew where your turns were all in one direction (either all to the right or all to the left). What is the sum of the measures of the exterior angles of each of these polygons? Is this sum the same for all polygons? Why or why not? **Note 7**

**Problem C4**

Use your drawings to determine the relationship between the measure of central angles and the measure of exterior angles in regular polygons. Explain this relationship.

Try this with simpler shapes, such as an equilateral triangle.

Video SegmentWatch this segment to see Mary and Susan experiment with Geo-Logo commands to create different regular polygons. They examine the relationship between the angle of turn and exterior and interior angles.
Do you think computer technology can enhance exploring mathematical tasks such as this one? Why or why not? You can find this segment on the session video approximately 16 minutes and 47 seconds after the Annenberg Media logo. |

**Problem C5**

In order to draw a non-intersecting star polygon, you must direct your pencil to turn both right and left. Print a copy of the star polygons and measure the exterior angles. What is the sum of the exterior angles in a star polygon? Explain this result using the commands.

Think about how turning in both directions affects the resulting exterior angles.

**Note 7**

If you are working in groups, review and discuss Problem C3. For each of the polygons drawn in Problem C1, have group members add the amount of turns that were used to draw the shape and bring the turtle back to its starting position. Discuss the following questions: What is the total sum of the turtle turns for each polygon? (Or, in other words, what is the sum of the exterior angles of these polygons?) If, when drawing the polygons, the turtle had not gone forward but rather had stayed in one spot, what figure would the turtle have drawn?

**Problem C1**

- To form a square, try this: Repeat 4; Go Forward 4; Rotate Right 90; End Repeat.
- To form a non-square rectangle, try this: Repeat 2; Go Forward 5; Rotate Right 90; Go Forward 7; Rotate Right 90; End Repeat.
- To form a non-rectangular parallelogram, try this: Repeat 2; Go Forward 5; Rotate Right 60; Go Forward 7; Rotate Right 120; End Repeat.
- To form an equilateral triangle, try this: Repeat 3; Go Forward 5; Rotate Right 120; End Repeat.
- To form a regular pentagon, try this: Repeat 5; Go Forward 5; Rotate Right 72; End Repeat.
- To form a regular hexagon, try this: Repeat 6; Go Forward 5; Rotate Right 60; End Repeat.
- There are many different star polygons, but here’s one way to do it: Repeat 5; Go Forward 5; Rotate Right 144; End Repeat. The angle must be a multiple of the angle used to form a regular polygon of the same number of sides. If you wanted a star polygon without intersecting lines, try this: Repeat 5; Go Forward 5; Rotate Left 72; Go Forward 5; Rotate Right 144; End Repeat.
- To form a regular n-gon, theoretically, the commands would be as follows: Repeat n; Go Forward 5; Rotate Right (360/n); End Repeat.

**Problem C2**

- If the turn is x degrees, the resulting interior angle measure is (180° – x°).
- Answers will vary, but one important observation is that such turns measure exterior angles. In order to build a polygon with the correct interior angles, we must first subtract from 180 degrees to find the exterior angles.

**Problem C3**

The sum of the measures of the exterior angles is 360 degrees for all polygons drawn this way. One explanation for why this is true is that the cursor must make a complete circle and return to its original position, and there are 360 degrees in a circle.

**Problem C4**

In a regular polygon, the measure of each central angle is equal to the measure of each exterior angle:

Since the central angles total 360 degrees and the exterior angles total 360 degrees, each of these angles is also equal. (For non-regular polygons, these totals are still equal, but the individual angles are not.)

**Problem C5**

See commands in **Problem C1 (h)**. The sum of the exterior angles is a multiple of 360 degrees, since the cursor will go around in a circle more than once as it moves through the points of the star. Answers will vary, depending on the size of the star polygon (specifically, the number of times the cursor must go around in a circle), but will always be a multiple of 360 degrees.