# Sec. 3-5 The Polygon Angle-Sum Theorems

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Sec. 3-5 The Polygon Angle-Sum Theorems
Objectives: To classify Polygons To find the sums of the measures of the interior & exterior s of Polygons.

Polygon: A closed plane figure. w/ at least 3 sides (segments)
The sides only intersect at their endpoints Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.

Which of the following figures are polygons?
No yes No

Example 1: Name the 3 polygons
Top XSTU S T Bottom WVUX X U Big STUVWX V W

I. Classify Polygons by the number of sides it has.
3 4 5 6 7 8 9 10 12 n Name Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon N-gon Interior  Sum

II. Also classify polygons by their Shape
a) Convex Polygon – Has no diagonal w/ points outside the polygon. E A D B C b) Concave Polygon – Has at least one diagonal w/ points outside the polygon. * All polygons are convex unless stated otherwise.

III. Polygon Interior  sum
4 sides 2 Δs 2 • 180 = 360 5 sides 3 Δs 3 • 180 = 540

S = (n -2) 180 All interior  sums are multiple of 180°
6 sides 4 Δs 4 • 180 = 720 All interior  sums are multiple of 180° Th(3-9) Polygon Angle – Sum Thm S = (n -2) 180 Sum of Interior  # of sides

Examples 2 & 3: Find the interior  sum of a 15 – gon. S = (n – 2)180
Find the number of sides of a polygon if it has an  sum of 900°. S = (n – 2)180 900 = (n – 2)180 5 = n – 2 n = 7 sides

Special Polygons: Equilateral Polygon – All sides are .
Equiangular Polygon – All s are . Regular Polygon – Both Equilateral & Equiangular.

IV. Exterior s of a polygon.
1 3 2 1 2 3 4 5

Th(3-10) Polygon Exterior  -Sum Thm
The sum of the measures of the exterior s of a polygon is 360°. Only one exterior  per vertex. 1 2 3 m1 + m2 + m3 + m4 + m5 = 360 5 4 For Regular Polygons The interior  & the exterior  are Supplementary. = measure of one exterior  Int + Ext = 180

Example 4: How many sides does a polygon have if it has an exterior  measure of 36°. = 36 360 = 36n 10 = n

Example 5: S = (n - 2)180 = (15 – 2)180 = 24 = (13)180 = 2340 n = 15
Find the sum of the interior s of a polygon if it has one exterior  measure of 24. S = (n - 2)180 = (15 – 2)180 = (13)180 = 2340 = 24 n = 15

Example 6: Solve for x in the following example. 4 sides
Total sum of interior s = 360 100 x = 360 280 + x = 360 x = 80

Example 7: Find the measure of one interior  of a regular hexagon. S = (n – 2)180 = (6 – 2)180 = (4)180 = 720 = 120

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