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Exploring Angles in Polygons: Theorems, Formulas, and Examples

Discover the properties of polygons with equal sides and angles, learn about diagonals, interior and exterior angles, and solve angle measurements with helpful formulas and examples.

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Exploring Angles in Polygons: Theorems, Formulas, and Examples

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  1. 6-1 Angles of a Polygon

  2. POLYGON: A MANY ANGLED SHAPE # sides = # angles = #vertices

  3. Some Info: • Regular Polygon: all angles are equal • Diagonal: a segment connecting 2 nonconsecutive vertices.

  4. DIAGONALS (Look at these, don’t write in notes) • Quadrilateral • Look! 2 triangles • 2(180) = 360 • Sum of the angles of a quadrilateral is 360 • Pentagon • 3 triangles • 3(180) = 540 • Sum of the angles of a pentagon is 540 • What do you think about a hexagon? • 4(180) = 720 • SO . . . . . . . .

  5. Theorem • The sum of the measures of the INTERIOR angles with n sides is (n – 2)180 • The sum of the measures of the exterior angles of any polygon is 360. • ALWAYS 360!!

  6. TWAP—(Try with a Partner) Hint: Just Plug into the formula! • Find a) the sum of the interior angles and b) the sum of the exterior angles for each shape 1) 32-gon 2) Decagon Answers: 1)a) 5400 b) 360 2)a) 1440 b) 360

  7. Other Formulas… The measure of EACH EXTERIOR angle of a regular polygon is: 360 n (It’s 360 divided by the number of sides) The measure of EACH INTERIOR angle of a polygon is: (n-2)180 n (It’s the SUM of Interior divided by # of sides)

  8. Example Find the measure of EACH interior angle of a polygon with 5 sides. (5-2)180 5 3(180)=540 540/5 = 108

  9. Since the sum of the measures of the interior angles is Example Find the measure of each interior angle of parallelogram RSTU. Step 1 Find the sum of the degrees!

  10. Example cont. Sum of measures of interior angles

  11. Example cont Step 2 Use the value of x to find the measure of each angle. mR = 5x = 5(11)= 55 mS = 11x + 4 = 11(11) + 4 = 125 mT = 5x = 5(11)= 55 mU = 11x + 4 = 11(11) + 4 = 125 Answer:mR = 55, mS = 125, mT = 55, mU = 125

  12. To Find # of sides… Formula: ____360____ 1 ext. angle (360 divided by 1 ext angle) Also: 1 interior angle + 1 exterior angle = 180

  13. Example How many sides does a regular polygon have if each exterior angle measures 45º? 360 45 n = 8 sides

  14. Example How many sides does a regular polygon have if each interior angle measures 120º? Find ext angle: 180-120= 60 360 60 n = 6 sides

  15. Example Find the value of x in the diagram.

  16. How many degrees will it =? 5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) + (5x + 5) = 360 (5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 + (–12) + 3 + 5] = 360 31x – 12 = 360 31x = 372 x = 12 Answer: x = 12

  17. Equations to Know (Flashcards!!!!) • Sum of interior angles • Each interior angle • Sum of exterior angles • Each exterior angle • # of Sides

  18. Homework • Pg. 398 #13-37 odd, 49

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