3.4 the polygon angle-sum theorems chapter 3: parallel and perpendicular lines
TRANSCRIPT
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3.4 The Polygon Angle-Sum Theorems
Chapter 3: Parallel and Perpendicular Lines
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3.4 The Polygon Angle-Sum Theorems
Polygon: a closed plane figure with at least three sides that are segments
A polygon Not a polygon;Not enclosed
Not a polygon;Two sides intersect
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Naming a Polygon
Name a polygon by its vertices.
A
B
CD
E
ABCDE or AEDCB
Start at one vertex and go around in order
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Naming a Polygon
Three polygons are pictured. Name each polygon:
L
M
NO
P
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Classifying a Polygon by the number of sides:
Sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n-gon
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Convex vs. Concave
A Convex Polygon has all vertices pointing “out”
A Concave Polygon has one or more vertices
“caving in”
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Classify
Classify each polygon by its sides. Identify each as convex or concave:
Hexagon; Convex Octagon; Concave
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Sum of Polygon Angle Measures
Use triangles to figure out the sum of the angles in each polygon:
# of Sides: # of Triangles:Total Degrees:
# of Sides:# of Triangles:Total Degrees:
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Sum of Polygon Angle Measures
Number of Sides Number of Triangles
Total Degrees inside Polygon
3 1 180
4
5
6
n
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Theorem 3-9 Polygon Angle Sum Theorem
The sum of the measures of the angles in a polygon
is (n – 2)180.
Find the sum of the measure of the angles of a 15-gon.
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Polygon Angle Sum
The sum of the measures of the angles of a given polygon is 720. How many sides does the polygon have?
Use (n – 2)180 :
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Using Polygon Angle-Sum Theorem
Find the measure of <Y in pentagon TVYMR at the right.
135°
M
Y V
TR
90°
Use (n – 2)180
Write an equation to solve for <Y
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Using Polygon Angle-Sum Theorem
Pentagon ABCDE has 5 congruent angles. Find the measure of each angle.
Use the Polygon Angle-Sum Theorem: (n – 2)180
Divide the total number of degrees by the number of angles:
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Exterior Angles
What do you notice about each set of exterior angles?
130°
150°
80°
115°
71°
75°
99°
86°
70°
88°
70°46°
1 2
31:
2:
3:
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Theorem 3-10 Polygon Angle-Sum Theorem
The sum of one set of exterior angles for any polygon is 360°.
1
2
3
4
5
m<1 + m<2 + m<3 + m<4 + m<5 = 360°
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Polygons
Equilateral Polygon: all sides congruent Equiangular Polygon: all angles congruent Regular Polygon: all sides and all angles congruent
(equiangular and equilateral)
*If a polygon is a regular polygon then all of the exterior angles are also congruent.
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Homework
Pg 147 1-25, 40-44, 47-49