Download 3-4 THE POLYGON ANGLE-SUM THEOREMS (p. 143

3-4 THE POLYGON ANGLE-SUM THEOREMS (p. 143-150)
A polygon is a closed plane (flat) figure whose sides are segments. The sides intersect
only at their endpoints, and no consecutive sides are collinear.
Example: Why are the following figures not polygons?
A
B
H
F
G
E
C
D
To name a polygon, start at any vertex and proceed consecutively in either a clockwise or
counterclockwise direction. Just do not skip letters along the way.
Example: Give at least two different names for the following polygon. Then, identify
its vertices, sides, and angles.
W
B
F
P
G
D
Do 1 on p. 143.
You can classify a polygon by the number of sides it contains.
Number of Sides
3
4
5
6
7
8
9
10
12
n
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
There are two types of polygons.
A convex polygon has no diagonal with points outside the figure.
Example: Sketch a convex polygon. Each interior angle is less than what measure?
A concave polygon has at least one diagonal with points outside the figure.
Example: Sketch a concave polygon. At least one interior angle has a measure greater
than what number?
We will focus primarily on convex polygons in this course.
Do 2a-b on p. 144.
Do the Investigation: The Sum of Polygon Angle Measures on p. 145. Answer 1-2.
Theorem 3-9 Polygon Angle-Sum Theorem
The sum of the measures of the (interior) angles of an n-gon is (n - 2)180.
Example: Find the sum of the measures of the angles of a nonagon. If the nonagon is
equiangular, what is the measure of each interior angle?
Do 3b on p. 145.
Example: Find
right angle.
m
X in quadrilateral WXYZ if X  Y, m W  102, and Z is a
You can draw an exterior angle at any vertex of a polygon by extending a side.
Do 46 a-c on p. 148 to prove the next theorem.
Theorem 3-10 Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is
360.
Example: Find the exterior angle-sum of a 52-gon.
An equilateral polygon has all of its sides congruent.
An equiangular polygon has all of its angles congruent.
A regular polygon is both equilateral and equiangular.
Example: Find the exterior angle-sum of a pentadecagon (a 15-sided polygon). If the
pentadecagon is regular, what is the measure of each exterior angle? Of each interior
angle?
Do 5 on p. 146. You will first need to read the packaging problem above it.
Homework p. 147-150: 3,7,9,14,16,21,24,26,31,34,36b,39,42,49-51,56,73,75,79,86