Transcript Polygon Angle Sum Conjectures

8.1 Find Angle Measures in Polygons
Objectives:
1. To find the sum of the measures of the
interior and exterior angles in any n-gon
Example 1
What is the sum of the interior angles in the
polygon below?
U
M
S
Example 2
What’s the difference between convex and
concave polygons?
Investigation 1
Using the two previous concepts, we will
discover a method for finding the sum of
the angles in any convex n-gon, where n is
the number of sides (or angles) of a given
polygon.
Step 1: Draw a series of convex n-gons,
starting with n = 3 and ending with n = 6.
Investigation 1
180
180
180
180
180
180
180
180
180
180
Step 2: In each polygon, draw all of the
diagonals from one vertex. Notice how
these diagonals divide the polygons into
triangles. How could this help find the sum
of the angles in each n-gon?
Investigation 1
180
180
180
180
180
180
180
180
180
180
Step 3: Complete the table.
Number of
sides
Number of
triangles
Angle Sum
3
4
5
6
Investigation 1
180
180
180
180
180
180
180
180
180
180
Step 4: Find a formula.
Number of
sides
3
4
5
6
Number of
triangles
1
2
3
4
Angle Sum
180°
360°
540°
720°
Polygon Interior Angles Theorem
The sum of the measures of the interior
angles of a convex n-gon is (n – 2)·180°.
m1 + m2 + … + mn = (n – 2)·180°
Example 3
What is the sum of the measures of the
interior angles of a convex octagon?
Example 4
What is the measure of each angle of an
equiangular octagon?
Example 5
Find the values of e and f.
Example 6
What is the measure of each angle in any
equiangular n-gon?
Equiangular Polygon Conjecture
The measure of each angle of an
equiangular n-gon can be found by using
either of the following expressions:
(n  2)  180
360
or 180 
n
n
Example 7
In a regular polygon, the measure of each
angle is 150. How many sides does the
polygon have?
Example 8: SAT
If the degree measures of the angles of a
quadrilateral are 4x, 7x, 9x, and 10x, what
is the sum of the measures of the smallest
angle and the largest angle?
Investigation 2
When you extend one
side of a triangle, you
form an exterior
angle. If you extend
each side of a
polygon to form one
exterior angle at each
vertex, you create a
set of exterior angles
for the polygon.
Investigation 2
Use the GSP activity
to investigate the
sum of the
measures of a set
of exterior angles of
a polygon.
Polygon Exterior Angles Theorem
The sum of the measures of one set of
exterior angles of a polygon is 360°.
Example 9
What is the value of x?
Example 10
What is the number of sides of a polygon in
which the sum of the degree measures of
the interior angles is 4 times the sum of the
degree measures of the exterior angles?
Example 11
What is the measure of each exterior angle
in an equiangular octagon?
What is the measure of each exterior angle
in an equiangular n-gon? How does this
relate to the Equiangular Polygon
Conjecture?
Assignment
• P. 510-513: 1, 2-10
even, 11-21, 24-28,
37-41
• Inscribed Polygons
Worksheet