Download (n – 2)(180) Polygon Angle-Sum Theorem

Section 3-4 Polygon Angle-Sum Theorem
SPI 32A: Identify properties of plane figures from information given in
a diagram
Objectives:
• Classify Polygons
• Find the sums of the measures of the interior and exterior
angle of polygons
Polygon:
• closed plane figure with at least 3 sides that are segments
• the sides intersect only at their endpoints
• no adjacent sides are collinear
Classify Polygons
Name Polygons By Their:
Vertices
Start at any vertex and list the vertices consecutively in a
clockwise direction (ABCDE or CDEAB, etc)
Sides
Name by line segment naming convention
AB, BC,CD, DE, EA
Angles
Name by angle naming convention
A, B, C, D, E
Classify Polygons by the Number of Sides
Number of Sides
Name
3
4
5
Triangle
Quadrilateral
Pentagon
6
8
Hexagon
Octagon
9
10
12
n
Nonagon
Decagon
Dodecagon
n-gon
Classify Polygons as Convex or Concave
Convex Polygon
Has no diagonals with
points outside the
polygon
Concave Polygon
Has at least one diagonal
outside the polygon
Classify Polygons as Convex or Concave
Classify the polygon below by its sides. Identify it as convex
or concave.
Starting with any side, count the number
of sides clockwise around the figure.
Because the polygon has 12 sides, it is a
dodecagon.
Think of the polygon as a star. If you draw a
diagonal connecting two points of the star that are
next to each other,that diagonal lies outside the
polygon, so the dodecagon is concave.
Triangle Angle-Sum Theorem
1. Draw and cut out a triangle.
2. Number the angles and tear them off.
3. Place the angles adjacent to each other.
4. Compare your results with others. What do you
observe about the sum of the angles of a triangle?
Triangle Angle-Sum Theorem
The sum of the measures of the
angles of a triangle measure 180º.
Polygon Angle-Sum Theorem
Use the Triangle Angle-Sum Theorem to find the sum of the measures
of the angles of a polygon.
1. Sketch convex polygons with 4, 5, 6, 7, and 8 sides. Construct a
table to record your data in order to look for a pattern or rule to find the
sum of the measures of the angles of an n-gon.
2. Divide each polygon into triangles by drawing all diagonals that are
possible from one vertex.
3. Multiply the number of triangles by 180 to find the sum of the
measures of the angles of each polygon.
Polygon
Number sides
(n)
4
n
# of Triangles Sum of Interior angle
measures (___∙ 180= ___)
2
2 ∙ 180 = 360
(n - 2) ∙ 180
Polygon Angle-Sum Theorem
Theorem 3-9: Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon is
(n - 2) 180.
Find the sum of the measures of the angles of a decagon.
A decagon has 10 sides, so n = 10.
Sum = (n – 2)(180)
Polygon Angle-Sum Theorem
= (10 – 2)(180)
Substitute 10 for n.
= 8 • 180
Simplify.
= 1440
Polygon Angle-Sum Theorem
The sum of the measures of the angles of a given
polygon is 720. How can you use the Polygon AngleSum Theorem to find the number of sides in the
polygon?
Sum = (n – 2) 180
Write the Equation
720 = (n – 2) 180
Sub. In known values
720 = 180n – 360
Simplify
1080 = 180n
Addition Prop of EQ
6=n
Hexagon (6 sides)
Use the Polygon Angle-Sum Theorem
Find m  X in quadrilateral XYZW.
The figure has 4 sides, so n = 4.
m
X + m Y + m Z + m W = (4 – 2)(180)
m X + m Y + 90 + 100 = 360
m
X+m
m
m
Y + 190 = 360
X+m
X+m
2m
m
Y = 170
X = 170
X = 170
X = 85
Polygon Angle-Sum Theorem
Substitute.
Simplify.
Subtract 190 from each side.
Substitute m X for m Y.
Simplify.
Divide each side by 2.
Polygon Exterior Angle-Sum Theorem
Equilateral Polygon:
• all sides are congruent
Equiangular Polygon:
• all angles are congruent
Regular Polygon:
• is both equilateral and equiangular
Real-world Connection
Below is a regular hexagon game board packaged in a
rectangular box. Explain how you know that all the
angles labeled 1 have equal measures.
The hexagon is regular, so all its angles are congruent.
An exterior angle is the supplement of a polygon’s angle
because they are adjacent angles that form a straight angle.
Because supplements of congruent angles are congruent,
all the angles marked 1 have equal measures.