Angles of Polygons Find the sum of the measures of the interior angles of a polygon Find the sum of the measures of the exterior angles of a This scallop.

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Angles of Polygons Find the sum of the measures of the interior angles of a polygon Find the sum of the measures of the exterior angles of a This scallop resembles a 12-sided polygon with diagonals drawn from one of the vertices.

Quadrilateral SUM OF MEASURES OF INTERIOR ANGLES Polygons with more than 3 sides have diagonals. Quadrilateral

Pentagon SUM OF MEASURES OF INTERIOR ANGLES Polygons with more than 3 sides have diagonals. Pentagon

Hexagon SUM OF MEASURES OF INTERIOR ANGLES Polygons with more than 3 sides have diagonals. Hexagon

Heptagon SUM OF MEASURES OF INTERIOR ANGLES Polygons with more than 3 sides have diagonals. Heptagon

Octagon SUM OF MEASURES OF INTERIOR ANGLES Polygons with more than 3 sides have diagonals. Octagon

Interior Angle Sum Theorem If a convex polygon has n sides then the sum S of the measures of its interior angles is: S = 180(n - 2)

EXAMPLE 1 N = 5 S = 180(n – 2) = 180(5 – 2) or 540 Find the sum of the interior angles of the pentagon. N = 5 S = 180(n – 2) = 180(5 – 2) or 540

Convex Polygons No. of sides n Name Angle Sum Sum ÷ n 3 triangle 180° 60° 4 quadrilateral 360° 90° 5 pentagon 540° 108° 6 hexagon 720° 120° 7 heptagon 900° 129° 8 octagon 1080° 135° 9 nonagon 1260° 140° 10 decagon 1440° 144° What is the exterior angle of each regular polygon? Is the total 360°in each case?

Interior Angles of Polygons Find the unknown angles below. x 100° w 90° 75° 120° 120° 70° 75° (5 – 2) x 180° = 540° 540 – = (4 – 2) x 180° = 360° 360 – 245 = 115° 125o 130o 136o z 100o 125o 112o 136o 108o 122o y 134o 126o (6 – 2) x 180° = 720° 720 – = (7 – 2) x 180° =

Interior Angles of Polygons Calculate the angle sum and interior angle of each of these regular polygons. 1 2 3 4 7 sides 9 sides 10 sides 11 sides Septagon/Heptagon Nonagon Decagon Hendecagon 900°/128.6° 5 6 7 12 sides 16 sides 20 sides Dodecagon Hexadecagon Icosagon

EXAMPLE 2 Find the measure of each interior angle n = 4 B C 2x° 2x° Sum of interior angles is 180(4 – 2) or 360 x° x° A D

Exterior Angles of Polygons Exterior Angle Theorem The exterior angle of a triangle is equal to the sum of the remote interior angles. remote interior angles A exterior angle B C D i.e. ACD = ABC + BAC

Exterior Angle Sum Theorem If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360° 2 1 3 5 4