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Geometry

Geometry. 3.5 Angles of a Polygon. Polygons (“many angles”). have vertices, sides, angles, and exterior angles are named by listing consecutive vertices in order. A. B. C. F. Hexagon ABCDEF. D. E. Polygons. formed by line segments, no curves the segments enclose space

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Geometry

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  1. Geometry 3.5 Angles of a Polygon

  2. Polygons (“many angles”) • have vertices, sides, angles, and exterior angles • are named by listing consecutive vertices in order A B C F Hexagon ABCDEF D E

  3. Polygons • formed by line segments, no curves • the segments enclose space • each segment intersects two other segments

  4. Polygons Not Polygons

  5. Diagonal of a Polygon A segment connecting two nonconsecutive vertices Diagonals

  6. Convex Polygons • No side ”collapses” in toward the center Easy test : RUBBER BAND stretched around the figure would have the same shape…….

  7. Convex Polygons Nonconvex Polygons

  8. From now on……. When the textbook refers to polygons, it means convex polygons

  9. Polygons are classified by number of sides Number of sidesName of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 8 octagon 10 decagon n n-gon

  10. Interior Angles of a Polygon • To find the sum of angle measures, divide the polygon into triangles • Draw diagonals from just one vertex 4 sides, 2 triangles Angle sum = 2 (180) 5 sides, 3 triangles Angle sum = 3 (180) 6 sides, 4 triangles Angle sum = 4 (180) DO YOU SEE A PATTERN ?

  11. Interior Angles of a Polygon 4 sides, 2 triangles Angle sum = 2 (180) 5 sides, 3 triangles Angle sum = 3 (180) 6 sides, 4 triangles Angle sum = 4 (180) The pattern is: ANGLE SUM=(Number of sides–2)(180)

  12. Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180. 5 sides. 3 triangles. Sum of angle measures is (5-2)(180) = 3(180) = 540 Example:

  13. Exterior Angles of a Polygon 3 2 2 1 4 3 5 1 4 5 Put them together The sum = 360 Works with every polygon! Draw the exterior angles

  14. Theorem The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.

  15. If a polygon is both equilateral and equiangular it is called a regular polygon Regular Polygons 120 120 120 120 120 120 120 120 120 120 120 120 Equilateral Equiangular Regular

  16. Example 1 • A polygon has 8 sides (octagon.) Find: • The interior angle sum • The exterior angle sum n=8, so (8-2)180 = 6(180) = 1080 360

  17. Example 2 Find the measure of each interior and exterior angle of a regular pentagon Interior: (5-2)180 = 3(180) = 540 540 = 108 each 5 Exterior: 360 = 72 each 5

  18. Example 3 • How many sides does a regular polygon have if: • the measure of each exterior angle is 45 • 360 = 45 360 = 45n • n n = 8 8 sides: an octagon • the measure of each interior angle is 150 • (n-2)180 = 150 (n-2)180 = 150n • n 180n – 360 = 150n • - 360 = - 30n • n = 12 12 sides

  19. In summary… • Sum of interior angles • (n-2)180 • Sum of ext. angles • 360 • One ext. angle • 360/n • One int. angle • [(n – 2)180]/n OR supp. to 360/n • # of sides given an ext. angle • 360/measue of ext. angle • # of sides given an int. angle • find the ext angle(supp to int. angle) • 360/measure of ext. angle

  20. Homework pg. 104 #1-17, skip 7, bring compass

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