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Sec. 3-5Sec. 3-5The Polygon Angle-Sum The Polygon Angle-Sum
TheoremsTheorems
Objectives: Objectives: a)a) To classify PolygonsTo classify Polygonsb)b) To find the sums of the To find the sums of the
measures of the interior & measures of the interior & exterior exterior s of Polygons.s of Polygons.
Polygon:Polygon:
A closed plane figure.A closed plane figure. w/ at least 3 sides (segments)w/ at least 3 sides (segments) The sides only intersect at their The sides only intersect at their
endpointsendpoints Name it by starting at a vertex & go Name it by starting at a vertex & go
around the figure clockwise or around the figure clockwise or counterclockwise listing each vertex counterclockwise listing each vertex you come across.you come across.
Which of the following figures are Which of the following figures are polygons?polygons?
yes No No
Example 1: Name the 3 polygons
S T
U
VW
X
Top
XSTU
Bottom
WVUX
Big
STUVWX
I. Classify Polygons by the number of sides it has.Sides
3
4
5
6
7
8
9
10
12
n
NameName
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
N-gon
Interior Interior SumSum
II. Also classify polygons by their Shape
a) Convex Polygon – Has no diagonal w/ points outside the polygon. E A
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
2 • 180 = 360
5 sides
3 Δs
3 • 180 = 540
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) 180S = (n -2) 180
Examples 2 & 3:Examples 2 & 3:
Find the interior Find the interior sum of a 15 – gon.sum of a 15 – gon.
S = (n – 2)180S = (n – 2)180
S = (15 – 2)180S = (15 – 2)180
S = (13)180S = (13)180
S = 2340S = 2340
Find the number of Find the number of sides of a polygon if sides of a polygon if it has an it has an sum of sum of 900°.900°.
S = (n – 2)180S = (n – 2)180
900 = (n – 2)180900 = (n – 2)180
5 = n – 2 5 = n – 2
n = 7 sides n = 7 sides
Special Polygons:Special Polygons:
Equilateral PolygonEquilateral Polygon – All sides are – All sides are ..
Equiangular PolygonEquiangular Polygon – All – All s are s are ..
Regular PolygonRegular Polygon – Both Equilateral & – Both Equilateral & Equiangular.Equiangular.
IV. Exterior s of a polygon.
1
23 1
2
3
4 5
Th(3-10) Polygon Exterior Th(3-10) Polygon Exterior -Sum -Sum ThmThm The sum of the The sum of the
measures of the measures of the exterior exterior s of a s of a polygon is 360°.polygon is 360°.
Only one exterior Only one exterior per vertex. per vertex.
1
2
3
4 5mm1 + m1 + m2 + m2 + m3 + m3 + m4 + 4 + mm5 = 3605 = 360
For Regular Polygons
360
n= measure of one exterior
The interior & the exterior are Supplementary. IntInt + Ext + Ext = = 180180
Example 4: Example 4:
How many sides does a polygon have How many sides does a polygon have if it has an exterior if it has an exterior measure of 36°. measure of 36°.
= 36
360 = 36n
10 = n
Example 5:Example 5: Find the sum of the interior Find the sum of the interior s of a s of a
polygon if it has one exterior polygon if it has one exterior measure measure of 24.of 24.
360
n= 24
n = 15
S = (n - 2)180
= (15 – 2)180
= (13)180
= 2340
Example 6: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: Example 7:
Find the measure of one interior Find the measure of one interior of of a a regular regular hexagon.hexagon.
S = (n – 2)180S = (n – 2)180
= (6 – 2)180= (6 – 2)180
= (4)180= (4)180
= 720= 720
720
6
= 120