 Ch 6.1 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.

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Ch 6.1 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.

Which of the following figures are polygons? yesNo

Special Polygons: Equilateral Polygon – Equilateral Polygon – All sides are . All sides are .

Special Polygons: Equiangular Polygon – Equiangular Polygon – All  s are . All  s are .

Special Polygons: Regular Polygon – Regular Polygon – Both Equilateral & Equiangular. Both Equilateral & Equiangular.

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

How many degrees are in a triangle? We know this by the Triangle Angle- Sum Theorem

III. Polygon Interior  sum A rectangle has how many sides? Without crossing lines, how many triangles can I make? Each triangle has 180 0, so if I have two triangles I have how many degrees?

III. Polygon Interior  sum How many sides does this polygon have? Without crossing lines, how many triangles can I make? Each triangle has 180 0, so if I have three triangles I have how many degrees?

6 sides 4 Δs 4 180 = 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. 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°. 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

IV. Exterior  s of a polygon. 1 23 1 2 3 45

Th(3-10) Polygon Exterior  -Sum Thm The sum of the measures of the exterior  s of a polygon is 360°. The sum of the measures of the exterior  s of a polygon is 360°. Only one exterior  per vertex. Only one exterior  per vertex. 1 2 3 4 5 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°. How many sides does a polygon have if it has an exterior  measure of 36°. = 36 360 = 36n 10 = n

Example 5: Find the sum of the interior  s of a polygon if it has one exterior  measure of 24. 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. Solve for x in the following example. x 100 4 sides Total sum of interior  s = 360 90 + 90 + 100 + x = 360 280 + x = 360 x = 80

Example 7: Find the measure of one interior  of a regular hexagon. 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

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