Download Section 6-1 Properties of Polygons

Section 6-1
Properties of Polygons
Classifying Polygons
Polygon: Closed plane figure with at least three sides that are
segments intersecting only at their endpoints, and no adjacent
sides are collinear.
B
B
C
A
A
B
C
C
A
E
D
Not a polygon
D
E
Not a polygon
D
E
Polygon
Classifying Polygons
Note: A diagonal of a polygon is a segments that connects
two nonconsecutive vertices.
Convex Polygon: A polygon with no diagonal with points
outside the polygon
Concave Polygon: A polygon with at least one diagonal with
points outside the polygon.
Convex
Concave
Classifying Polygons – Example 1
Classify each polygon by its sides. Identify each as
convex or concave.
Convex Octogon
Concave Hexagon
Concave 20-gon
Classifying Polygons
Equilateral Polygon: A polygon
with all sides congruent.
Equiangular Polygon: A polygon
with all angles congruent.
Regular Polygon: A polygon that is both equilateral and
equiangular.
Classifying Polygons
We can classify polygons according to the number of sides it has.
Sides
Name
3
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
4
5
6
8
9
10
12
n
Classifying Polygons – Example 2
Tell whether the polygon is regular or irregular. Tell
whether it is concave or convex.
Regular, Convex
Irregular; Concave
Regular, Convex
Naming Polygons
D
C
G
H
F
K
J
•To name a polygon, start at any vertex and
list the vertices consecutively in a clockwise or
counterclockwise direction.
Polygon:
DGHJKFC or GDCFKJH
Vertices:
C,D,G,H,J,K,F
Sides:
Angles:
DG, GH, HJ, JK, KF, FC, CD
<D, <G, <H, <J, <K, <F, <C
Class Work…  Complete the table with a partner/ group
Theorem 3-14: Polygon Angle-Sum Theorem: The sum of the
measures of the angles in an n-gon is (n – 2)180.
Assignment #39
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AND
DEFINE TERMS: Polygon, Convex Polygon, Concave Polygon,
Equilateral Polygon, Equiangular Polygon, Regular Polygon, &
Polygon Angle Sum Theorem
Section 6-1 (cont.)
Properties of Polygons
Polygon Angle Sum Theorem
Theorem 3-14: Polygon Angle-Sum Theorem: The sum of the
measures of the angles in an n-gon is (n – 2)180.
Polygon Angle Sum – Example 3
Find the sum of the measures of the angles of a regular
dodecagon. Then find the measure of each interior angle.
= (n – 2)180
Polygon Angle Sum Theorem
= (12 – 2)180
Substitution
= (10)180
Simplify
= 1800
Simplify
**Be Careful!! Are they asking for the SUM of the angles,
or for EACH angle measure?**
Example 4
The sum of the angles of a polygon is 720. Find the number
of sides the polygon has and classify it.
Sum of the Angles = (n – 2)180
720 = (n – 2)180
4 = (n – 2)
6=n
Hexagon
Polygon Angle Sum Theorem
Substitution
Divide both sides by 180
Add 2 to both sides
Example 5
Find x in the following polygon.
125°
x°
125°
To solve, we will use the polygon
angle sum theorem for n=5
90 + 90 + 125 + x + 125 = (5 – 2)180
Polygon Angle Sum Theorem
330 + x = (5 - 2)180
330 + x = 540
x = 210
Simplify
Simplify
Subtract 330 from both sides
Example 6
Find the measure of each angle in the following polygon.
To solve, we will use the polygon angle
sum theorem for n=5
(5 x  2)  (5 x  10)  (4 x  15)  (8 x  8)  (3x  5)  (5  2)(180)
Polygon Angle Sum Theorem
Simplify / Combine Like Terms
25 x  40  (3)(180)
Simplify
25x  40  540
Solve for x
25x  500
x  20
Now, solve for each angle in the polygon…
A  5x  2  5(20)  2 102
E  5x  10  5(20)  10 120
D  4x 15  4(20)  15
 95
C  8x  8  8(20)  8 168
 65
B  3x  5  3(20)  5
Polygon Angle Sums
Theorem 3-15: Polygon Exterior Angle-Sum Theorem: The
sum of the measures of the exterior angles of a polygon, one at
each vertex, is 360.
For a pentagon: m1 m2  m3  m4  m5  360
3

2
4
1
5
Polygon Angle Sums – Example 7
Find the measure of each exterior angle of a regular dodecagon.
**The sum of the exterior
angles is 360 degrees**
Set up an equation
**Remember, a regular polygon has
both equal sides and equal angles
Exterior Angle Sum= Measure of Each Angle (Number of Sides)
360 = x(12)
Now, Solve.
30 = x
Divide both sides by 12
Each exterior angle is 30 degrees
Example 8
Find the measure of each angle in the following polygon.
To solve, we will use the polygon exterior angle sum
theorem & set all angles equal to 360
33b 18b 15b  28b 10b 16b  360
Polygon Exterior Angle Sum Theorem
Simplify / Combine Like Terms
120b  360
Solve for b
b3
Now, solve for each angle in the polygon…
H  33b  33(3)  99
G  18b  18(3)  54
F  15b  15(3)  45
L  28b  28(3)  84
K  10b  10(3)  30
J  16b  16(3)  48
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