Download 6.1 Polygons - Teacher Notes

3.5 The Polygon Angle-Sum
Theorems
Geometry
Mr. Barnes
Objectives:
• To Classify Polygons
• To find the sums of the measures of the
interior and exterior angles of polygons.
Q
VERTEX
R
Definitions:
SIDE
P
S
VERTEX
T
• Polygon—a plane figure that meets the following
conditions:
– It is formed by 3 or more segments called sides, such that no two
sides with a common endpoint are collinear.
– Each side intersects exactly two other sides, one at each endpoint.
• Vertex – each endpoint of a side. Plural is vertices.
You can name a polygon by listing its vertices
consecutively.
For instance, PQRST and QPTSR are two
correct names for the polygon above.
Example 1: Identifying Polygons
State whether the
figure is a polygon.
If it is not, explain
why.
• Not D- because D has a
side that isn’t a segment –
it’s an arc.
• Not E- because two of
the sides intersect only
one other side.
• Not F- because some of
its sides intersect more
than two sides.
A
C
B
F
E
D
Figures A, B, and C are polygons.
Polygons are named by the number
of sides they have – MEMORIZE
Number of Sides
Type of Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
Polygons are named by the number
of sides they have – MEMORIZE
Number of sides
Type of Polygon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n-gon
Convex or concave?
• Convex if no line that
contains a side of the
polygon contains a point
in the interior of the
polygon.
• Concave or non-convex
if a line does contain a
side of the polygon
containing a point on the
interior of the polygon.
See how this crosses
a point on the inside?
Concave.
See how it doesn’t go on the
Inside-- convex
Convex or concave?
• Identify the polygon
and state whether it is
convex or concave.
A polygon is EQUILATERAL
If all of its sides are congruent.
A polygon is EQUIANGULAR
if all of its interior angles are
congruent.
A polygon is REGULAR if it is
equilateral and equiangular.
P
Ex. : Interior Angles of a
Quadrilateral
80°
70°
x°
x°+ 2x° + 70° + 80° = 360°
3x + 150 = 360
3x = 210
x = 70
Q
2x°
R
• Sum of the measures of int. s
of
• A quadrilateral is 360°
• Combine like terms
• Subtract 150 from each side.
• Divide each side by 3.
Find m Q and mR.
mQ = x° = 70°
mR = 2x°= 140°
►So, mQ = 70° and mR = 140°
S
Investigation Activity
• Sketch polygons with 4, 5,
6, 7, and 8 sides
• Divide Each Polygon into
triangles by drawing all
diagonals that are
possible from one vertex
• Multiply the number of
triangles by 180 to find
the sum of the measures
of the angles of each
polygon.
1) Look for a pattern.
Describe any that
you have found.
2) Write a rule for the
sum of the measures
of the angles of an
n-gon
Polygon Angle-Sum Theorem
• The sum of the
measures of the
angles of an n-gon is
(n-2)180
• Ex: Find the sum of
the measures of the
angles of a 15-gon
• Sum = (n-2)180
•
= (15-2)180
•
= (13)180
= 2340
Example
• The sum of the
interior angles of a
polygon is 9180. How
many sides does the
polygon have?
•
•
•
•
•
Sum = (n-2)180
9180 = (n-2)180
51 = n-2
53 = n
The polygon has 53
sides.
Polygon Exterior Angle-Sum
Theorem
• The sum of the
measures of the
exterior angles of a
polygon, one at each
vertex, is 360.
• An equilateral
polygon has all sides
congruent
• An equiangular
polygon has all angles
congruent
• A regular polygon is
both equilateral and
equiangular.