Objectives: Develop and use formulas for the sums of the
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Objectives:Develop and use formulas for the sums of the measures of interior and exterior angles of polygons
3.6 Angles in Polygons
Warm-Up: Here’s a two part puzzle designed to prove that half of eleven is six. First rearrange two sticks to reveal the number eleven. Then remove half of the sticks to reveal the number six.
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Convex Polygon:
A polygon in which any line segment connecting two points of the polygon has no part outside the polygon.
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Concave Polygon:
A polygon that is not convex.
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Consider the following Pentagon:
𝟏Divide the polygon into three triangular regions by drawing all the possible diagonals from one vertex.
Find each of the following:
𝟐𝟑𝟒
𝟔
𝟓𝟗
𝟕𝟖
Add the three expressions:
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Note: You can form triangular regions by drawing all possible diagonals from a given vertex of any polygon
Polygon # of sides
# of triangular
regions
Sum of Interiorangles
Triangle
Quadrilateral
Pentagon
Hexagon
3
4
5
n-gon
6
n
1
2
3
4
n2
180
360
540
720
180(n-2)
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The sum of the measures of the interior angles of a polygon with n sides is:
180(n-2)
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Note: Recall that a regular polygon is on in which all the angles are congruent.
Polygon # of sides
Measure of Interior
angles
Sum of Interiorangles
Triangle
Quadrilateral
Pentagon
Hexagon
3
4
5
n-gon
6
n
180
360
540
720
180(n-2)
60
90
108
120180(n-2)
n
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The measure of an Interior Angle of a Regular Polygon with n sides is:
180(n-2) n
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Exterior Angle Sums in Polygons
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Polygon # of sides
Sum of interior &exterior angles
Sum of Interiorangles
Triangle
Quadrilateral
Pentagon
Hexagon
3
4
5
n-gon
6
n 180n
180
720
540
720
180(n-2)
Sum of Exteriorangles
360
360
360
360
360540
360
900
1080
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Theorem
𝟑𝟔𝟎𝟎
Sum of the measures of the Exterior Angles of a Polygon is:
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For a Convex Polygon
For a Regular Polygon
Polygon
Numberof
Sides
Number of
ΔRegio
ns
Sum of Interior Angles
Sum of
Exterior
Angles
Sum of Int & Ext Angle
s
Measure of
Interior Angles
Measure of
Exterior Angles
Triangle 3 1 180 360 540 60 120Quadrilater
al4 2 360 360 720 90 90
Pentagon 5 3 540 360 900 108 72Hexagon 6 4 720 360 108
0120 60
Heptagon 7 5 900 360 1260
128.6 51.4
Octagon 8 6 1080 360 1440
135 45
Nonagon 9 7 1260 360 1620
140 40
Decagon 10 8 1440 360 1800
144 36
11-gon 11 9 1620 360 1980
147.3 32.7
Dodecagon 12 10 1800 360 2160
150 30
13-gon 13 11 1980 360 2340
152.3 27.2
n-gon n n-2 180(n-2)
360 180n
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Find the indicated angle measures.
𝟓𝟒𝟎
𝟒𝟖𝟎
𝒙𝟎
𝟏𝟎𝟎𝟎
𝟏𝟏𝟓𝟎𝟏𝟐𝟎𝟎
𝒛 𝟎
𝒚𝟎
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Find the indicated angle measures.
𝟗𝟎𝟎
𝒙𝟎
𝟕𝟓𝟎
𝟖𝟓𝟎 𝟏𝟑𝟐𝟎
𝟏𝟐𝟎𝟎
𝒚𝟎
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Find the indicated angle measures.
𝟗𝟎𝟎
𝒙𝟎 𝟏𝟎𝟎𝟎
𝟏𝟑𝟎𝟎
𝟏𝟏𝟎𝟎
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Find the indicated angle measures.
𝟏𝟎𝟓𝟎𝒙𝟎
𝟏𝟎𝟎𝟎
𝟏𝟏𝟎𝟎
𝟕𝟓𝟎
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For each polygon determine the measure of an interior angle and the measure of an exterior angle.
A rectangle
A regulardodecagon
An equilateraltriangle
An equiangularpentagon
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An interior angle measure of a regular polygon is given. Find the number of sides of the polygon
𝟏𝟑𝟓𝟎 𝟏𝟓𝟎𝟎 𝟏𝟔𝟓𝟎
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An exterior angle measure of a regular polygon is given. Find the number of sides of the polygon
𝟔𝟎𝟎 𝟑𝟔𝟎 𝟐𝟒𝟎
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Find the indicated angle measure.(𝟒 𝒙 )𝟎 (𝟑 𝒙)𝟎
(𝟐 𝒙)𝟎(𝒙)𝟎
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Find the indicated angle measure.¿¿ (𝟐 𝒙+𝟑𝟎)𝟎
(𝟖 𝒙−𝟏𝟎)𝟎
(𝒙 ¿¿𝟐+𝟏𝟎)𝟎¿
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Find the indicated angle measure.
¿¿
¿
¿¿
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