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B E L L R I N G E R

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. 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.

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B E L L R I N G E R

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  1. 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. What is the measure of one angle in a regular hexagon?

  2. SOLUTIONS • SOLUTION #1 • 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°. • 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) • So, the last angle would be 540 – 380 = 160°

  3. 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. • 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°.

  4. 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: 6 ∠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. 1 5 2 4 3

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

  6. 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°. • 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°

  7. 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° What about the measure of one angle in the 1000-gon? 179,640/1000 = 179.64° What is therefore the measure of one exterior angle? Since it is a linear pair to the interior angle, 180 – 179.64 = .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.

  8. 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) Since it’s a linear pair with the interior angle, simply subtract it from 180, and you have the INTERIOR angle! The measure of one exterior angle.

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

  10. Try This . . . x = ? y = ? First, you can find y because it is a linear pair with 50°. y + 50° = 180° -50 -50 y = 130° x x 135° y x y 50° 40° 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, 130 + 130 + 135 + 140 + x + x + x = 900. 535 + 3x = 900  3x = 365  x = 365/3 = 121.7°

  11. Or This . . . First, y and 116° are LP. 180 -116 = 64° All of the exterior angles must measure a total of 360°. Thus, 64 + 82 + x + x + x + 90 = 360° 236 + 3x = 360 -236 -236 3x = 124 x = 124 / 3 = 41.3° y 116° x 82° z x x Now find z. It is a Linear Pair with x which we found to be 41.3° So, z = 180 – 41.3 = 138.7°

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

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