B E L L R I N G E R Four angles of a pentagon are 60 , 110 , 136 , and 74 . Find the measure of the last angle. Four angles of a pentagon are 60 , 110 , 136 , and 74 . Find the measure of the last angle. What is the measure of one angle in a regular hexagon? What is the measure of one angle in a regular hexagon?

SOLUTIONS SOLUTION #1 In a pentagon, there are 5 sides, 5 vertices, and 5 angles. In a pentagon, there are 5 sides, 5 vertices, and 5 angles. We are given the measure of 4 angles, which in total measure: 60° + 110° + 136° + 74° = 380°. We are given the measure of 4 angles, which in total measure: 60° + 110° + 136° + 74° = 380°. The sum of all the interior angles in a pentagon can be found using (n-2)180. n=5, so (5-2)180 = 3(180) = 540°. (See example) The sum of all the interior angles in a pentagon can be found using (n-2)180. n=5, so (5-2)180 = 3(180) = 540°. (See example)example So, the last angle would be 540 – 380 = 160° So, the last angle would be 540 – 380 = 160°

SOLUTIONS SOLUTION #2 First, find the sum of all the angles inside of a hexagon. (n-2)180 = sum of angles inside of a polygon. First, find the sum of all the angles inside of a hexagon. (n-2)180 = sum of angles inside of a polygon. So, (6-2) 180 = (4) 180 = 720°. So, (6-2) 180 = (4) 180 = 720°. Since there are six angles in the hexagon, and since they must each be the same measure (the hexagon is regular), we can divide 720 by 6 or 720/6 = 120°. Since there are six angles in the hexagon, and since they must each be the same measure (the hexagon is regular), we can divide 720 by 6 or 720/6 = 120°.

More about POLYGONS: What about the angles EXTERIOR of the polygon? EXTERIOR angles of a polygon, are the ‘LINEAR PAIRS’ of the INTERIOR ANGLES. Observe below: ∠ 1, ∠ 2, ∠ 3, ∠ 4, ∠ 5, and ∠ 6 are all EXTERIOR angles of the hexagon. They each form a linear pair with the INTERIOR angles of the figure.

Polygon Exterior Angle Theorem For any polygon, the sum of the measures of the exterior angles will equal 360° OBSERVE in class investigation. investigation For example, 25°70° 85° 155° 95° 110° °

Example Suppose you are observing a regular nonagon. What is the measure of each exterior angle? We know the sum of the exterior angles for any polygon must be 360°. We know the sum of the exterior angles for any polygon must be 360°. The nonagon is regular—all the interior angles must be congruent. Therefore, all of the exterior angles are congruent. The nonagon is regular—all the interior angles must be congruent. Therefore, all of the exterior angles are congruent. There are 9 sides, 9 interior angles, and 9 exterior angles. 360/9 = 40° There are 9 sides, 9 interior angles, and 9 exterior angles. 360/9 = 40°

How about this? Suppose you have an ‘equiangular’ 1000-gon. What is the sum of all the angles in a 1000-gon? (1000-2)180 = 179,640° (1000-2)180 = 179,640° What about the measure of one angle in the 1000-gon? 179,640/1000 = ° What is therefore the measure of one exterior angle? Since it is a linear pair to the interior angle, 180 – =.360° So what is the measure of all the exterior angles in a 1000-gon? (.360)1000 = 360° --just like every other polygon that has ever existed.

Equiangular Polygon Conjecture In the Bell Ringer, we had found the measure of one angle inside of an equiangular polygon using our own formula— (n-2)180 / n (That is, the sum of the angles in the polygon, divided by the number of sides/angles in the polygon) Now we know a new way to find this information. For any equiangular polygon, the measure of just one angle in the polygon can be found by 180-(360/n) The measure of one exterior angle. Since it’s a linear pair with the interior angle, simply subtract it from 180, and you have the INTERIOR angle!

HUH? This means that: (n-2)180 = 180 – 360 Are these two really the same?? (n-2)180 = 180 – 360 Are these two really the same?? n n n n Let’s see– multiply both sides by ‘n’. (n-2)180 = 180n – n – 360 = 180n – 360 (distributive property) Both formulas are the same □

Try This... x x x y y 135° 40° 50° x = ? y = ? First, you can find y because it is a linear pair with 50°. y + 50° = 180° y = 130° Now, using polygon angle sum we can find x. One of the angles of this septagon we can find since it is a linear pair with 40°. 180 – 40 = 140° Now use Polygon angle sum. A septagon has 7 sides. (7-2)180. (5)180 = 900°. So, x + x + x = x = 900  3x = 365  x = 365/3 = 121.7°

Or This... y x x x 116° 82° z First, y and 116° are LP = 64° All of the exterior angles must measure a total of 360°. Thus, x + x + x + 90 = 360° x = x = 124 x = 124 / 3 = 41.3° Now find z. It is a Linear Pair with x which we found to be 41.3° So, z = 180 – 41.3 = 138.7°

CLASSWORK P.257,258, #3, 4, 5, 8, 10, 13 P. 262 #2-7 Due when you leave