Download 8.1Find Angle Measures in Polygons

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Vocab
Consecutive vertices: vertices that are next to each
other or part of the same side.
FIND ANGLE MEASURES IN
POLYGONS
Ch 8.1
Diagonal
A diagonal is a segment that connects 2 nonconsecutive edges.
Polygon Interior Angles Theorem
Theorem 8.1
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What is the sum of the interior angles
of…
(n − 2) • 180
A triangle:
Corollary to Interior Angle Theorem
(3 − 2) • 180 = 180
A quadrilateral:
(4 − 2) • 180 = 360
A pentagon:
(5 − 2) • 180 = 540
A hexagon:
(6 − 2) • 180 = 720
A heptagon:
(7 − 2) • 180 = 900
If I have the Sum of the interior angles how
do I figure out how many sides there are?
If I have the Sum of the interior angles how
do I figure out how many sides there are?
(n − 2) • 180 = sum of angles
Given that the sum is 1800
(n − 2) • 180 = 1800
(n − 2) • 180 1800
=
180
180
n − 2 = 10
n = 12
(n − 2) • 180 = sum of angles
Given that the sum is 1980
(n − 2) • 180 = 1980
(n − 2) • 180 1980
=
180
180
n − 2 = 11
n = 13
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How would you find x?
What kind of figure is this?
What is the sum of the interior
angles?
What kind of figure is this?
What is the sum of the interior
angles of a pentagon?
(5 – 2)(180) = 540
108 + 121 + 59 + x = 360
288 + x = 360
110+92+100+84 + x = 540
386 + x = 540
x = 154
x = 72°
What kind of figure is it?
How do you find x?
Finding exterior angles
What is the sum of the interior
angles of a heptagon?
(7 – 2)(180) = 900
145+112+99+133+156+2x+x = 900
645 + 3x = 900
3x = 255
x = 85
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How do you find x?
Find x
X+152+137=360
X+289=360
X = 71
67 + 96 + 59 + 86 + x = 360
308 + x = 360
x = 52
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Word Problems
What is the measure of one exterior angle in a regular
triangle?
120 ̊
Recall that the sum of any convex
polygon’s exterior angles is 360.
Then divide by how many exterior
angles there are in the shape.
360 ÷ 3 = 120
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What is the measure of an exterior
angle in a regular 10-gon?
360 ÷ 10 = 36 ̊
The measures of the exterior angles of
a convex pentagon are 54, 72, 2x, 3x
and x. What is the measure of the
largest angle?
The sum of the measure of the exterior angles
equals 360, so we’ll add the angles and find x.
54 + 72 + 2x + 3x + x = 360
126 + 6x = 360
6x = 234
X = 39
3x = 3(39) = 117
Find an interior and exterior angle for
a regular 11-gon
How do you find how many sides a figure
has, when given one angle measure?
360 ÷ 11 = 32.73 ̊
X + 32.73 = 180
X = 147.27
How do you find the sum of the interior angles?
(n – 2)180 = interior angle sum
If we have a regular figure all the angles are the
same, so we would divide the interior angle sum by
the number of sides, n, to find the measure of one
angle. (n − 2)180
n
= one angle measure
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An interior angle of a regular polygon has
a measure of 150 ̊. How many sides does
the figure have?(n − 2)180
n
= one angle measure
An interior angle of a regular polygon has
a measure of 120 ̊. How many sides does
the figure have?(n − 2)180
n
= one angle measure
( n − 2)180
= 150 Multiply both sides by n!
n
(n − 2)180
n⋅
= 150 ⋅ n
n
(n − 2)180 = 150 ⋅ n Distribute the 180!
( n − 2)180
= 120 Multiply both sides by n!
n
(n − 2)180
n⋅
= 120 ⋅ n
n
(n − 2)180 = 120n Distribute the 180!
180n − 360 = 150n Subtract the 180n!
− 360 = −30n
12 = n
180n − 360 = 120n Subtract the 180n!
− 360 = −60n
6=n
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