Download 1 - GEOMETRY 9

Find Angle Measures in Polygons (8.1)
Definition: A diagonal of a polygon is a segment that
joins 2 non-consecutive vertices
B
C
A
AD is a diagonal of ABCDE
D
E
Ex. Draw the non-intersecting diagonals of the polygon below.
**Notice that the diagonals divide the polygons into triangles.
Ex. How many triangles were formed? ________
**Remember by the Triangle-Angle-Sum Theorem the sum
of the angles of a triangle is equal to 180˚
Ex. What is the sum of the interior angles of the polygon above?
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Figure B
Figure A
Figure C
Figure D
Figure E
Complete the following Chart
Figure
Number of
Sides
Number of Sum of the angles
Δ’s Formed
of the polygon
A
B
C
D
E
any
2
Theorem 8.1
Polygon Interior Angles Theorem
The sum of the measure of the interior angles of a convex n-gon
is __________________
Corollary to Theorem 8.1 Interior Angles of a Quadrilateral
The sum of the measures of the interior
angles of a quadrilateral is _____________
2
1
3
4
m1 + m2 + m3 + m4 = __________
**Remember Polygons can be classified by the number of sides:
Number of Sides
3
4
5
6
7
8
9
12
Name of Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n
n-gon
10
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Ex. Find the sum of the measures of the interior angles of a convex
octagon.
Ex. Find the sum of the interior angles of a convex dodecagon.
S
R
Ex. Find the value of x
165˚
Y
115˚
95˚ T
160˚
x˚
80˚
W
130˚
U
V
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Ex. The measures of 3 interior angles of a quadrilateral are 89˚, 110˚, and
46˚. Find the measure of the fourth angle.
Ex. Find the missing angles
if mB = mE
B
85˚
A 93˚
156˚
C
D
E
Ex. Given the sum of the angles of a convex polygon is 1620º,
classify the polygon by the number of sides.
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Definition: Exterior Angle of a polygon is an angle formed by a
side and extension of a side.
4
4 is an exterior angle of the
triangle
3
1
2
Exterior Angle + Interior Angle = 180º
Theorem 8.2
Polygon Exterior Angles Theorem
The sum of the measures of the
exterior angles of ANY convex
polygon, one at each vertex,
is ALWAYS equal to 360˚
3
2
4
1
5
m1 + m2 + m3 + m4 + m5 = __________
Ex. Find the value of x
60˚
85˚
45˚
x˚
50˚
40˚
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Definition: A Regular Polygon has all sides congruent and all
angles congruent.
 The SUM of ALL the INTERIOR ANGLES of a
polygon = (n – 2)180
 The measure of EACH one of the INTERIOR
ANGLES of a REGULAR polygon
minterior = ________
 The SUM of ALL the EXTERIOR ANGLES of
a polygon = 360º
 The measure of EACH one of the EXTERIOR
ANGLES of a REGULAR polygon
mexterior = ________
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There are 2 ways to find an interior and exterior angle of a
regular polygon:
METHOD 1:
Step 1: Find the measure of one interior angle using:
m interior = (n – 2)180 ÷ n
Step 2: Find the measure of the exterior angle = 180 – m interior
Ex. Find the measure of each interior and exterior angle of a regular
pentagon
Interior Angle:
Exterior Angle:
METHOD 2:
Step 1: Find the measure of one exterior angle using m = 360 ÷ n
Step 2: Find the measure of the interior angle = 180 – m exterior
Ex. Find the measure of each angle of a regular octagon
Exterior Angle:
Interior Angle:
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