1. B E L L R I N G E R What is the measure of one exterior angle of a regular octagon? What is the measure of one interior angle of a regular decagon? How many sides does a regular polygon have if the measure of one exterior angle measures 72°?

2. SOLUTION 1  For ANY polygon, all of the exterior angles must add up to 360°.  This polygon is an octagon—it must have 8 interior angles and 8 exterior angles.  Since the octagon is regular, all 8 exterior angles are congruent.  360/8 = 45°

3. SOLUTION 2  We learned last class session that one angle measure in an equiangular polygon = 180 – (360/n)  A decagon has 10 sides, therefore, 180 – (360/10) = 180 – (360/10) = 180 – 36 = 180 – 36 = 144° 144° Don’t forget our old way of solving this problem, which is just as valid: (n-2)180 n

4. SOLUTION 3  We are told that one exterior angle equals 72°.  We are told the polygon is regular, thus each exterior angle must be 72°.  But, every polygon’s exterior angles may only add up to 360°.  360/72 = 5 exterior angles  5 sides, or  PENTAGON.

5. Additional Examples  If the sum of all the interior angles of a polygon is 2,340°, then how many sides must the polygon have? The sum of all the angles in a polygon can be found by (n – 2)180. The sum of all the angles in a polygon can be found by (n – 2)180. Thus, (n – 2)180 = 2,340 Thus, (n – 2)180 = 2,340 180n – 360 = 2,340 180n – 360 = 2,340 +360 +360 +360 +360 180n = 2,700  2,700/180 = n = 15 sides. 180n = 2,700  2,700/180 = n = 15 sides.

6. Examples –cnt’d-  An exterior angle of a regular polygon measures 18°. What is the measure of an interior angle?  What is the sum of all the interior angles? First, remember that exterior angles and interior angles are linear pairs. An interior angle measures 180 – 18 = 162°. First, remember that exterior angles and interior angles are linear pairs. An interior angle measures 180 – 18 = 162°. Next, let’s determine how many sides there are. Since one exterior angle is 18°, and every polygon’s exterior angles add up to 360°, then 360/18 = 20 exterior angles  20 sides. Next, let’s determine how many sides there are. Since one exterior angle is 18°, and every polygon’s exterior angles add up to 360°, then 360/18 = 20 exterior angles  20 sides. Therefore, there are 20 interior angles and each measures 162°  (20)(162) = 3,240°. Therefore, there are 20 interior angles and each measures 162°  (20)(162) = 3,240°.