Sec. 3-5 The Polygon Angle-Sum Theorems Objectives: a)To classify Polygons b)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? yesNo

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

I. Classify Polygons by the number of sides it has. Sides nName 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. EA B C D 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 = sides 3 Δs = 540

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

Examples 2 & 3:  Find the interior  sum of a 15 – gon. S = (n – 2)180 S = (15 – 2)180 S = (13)180 S = 2340  Find the number of sides of a polygon if it has an  sum of 900°. S = (n – 2) = (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

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 m  1 + m  2 + m  3 + m  4 + m  5 = 360 For Regular Polygons = measure of one exterior  The interior  & the exterior  are Supplementary. Int  + Ext  = 180

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

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

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

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