Download Chapter 1 Points, Lines, Planes, and Angles page 1

Lesson 3-5
Angles of a Polygon
(page 101)
Essential Question
How can you apply parallel
lines (planes) to make
deductions?
POLYGON:
A plane figure formed by
(sides) such that:
coplanar
(1) each segment intersects exactly
segments
two
other
segments, one at each endpoint; and …
(2) no two segments with a common endpoint
are
collinear .
Polygon means “many
angles .”
Examples of Polygons
A
B
E
D
C
Written: polygon ABCDE or ABCDE
(write vertices in consecutive order)
Example of a Polygon
Example of a Figure that
is not a Polygon
Example of a Figure that
is not a Polygon
CONVEX POLYGON:
A polygon such that no line containing a side of the
polygon contains a point in the interior of the polygon.
Nonconvex Polygons - examples
DIAGONAL:
A segment joining two non - consecutive vertices of a polygon.
Nonconvex Polygons will have a diagonal in the exterior.
Activity: Draw all the diagonals from one vertex
on each polygon, then complete the chart.
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
180º
# of sides of
polygon
3
4
5
6
7
8
9
10
11
12
n
Name of Polygon
triangle
# of diagonals
from 1 vertex
# of triangles
formed
sum of angle
measures
0
1
180º
1
2
360º
pentagon
2
3
540º
hexagon
3
4
720º
septagon
4
5
900º
octagon
5
6
1080º
nonagon
6
7
1260º
decagon
7
8
1440º
undecagon
8
9
1620º
dodecagon
9
10
1800º
n-3
n-2
quadrilateral
n-gon
(n-2)180º
Theorem 3-13
The sum of the measures of the angles of a convex
polygon with n sides is
(n - 2)  180º .
Theorem 3-14
The sum of the measures of the exterior angles of any
convex polygon, one angle at each vertex, is
360º .
Refer to page 97, #10 and page 104, #7 to verify this thm.
Paper Polygon Proof
We will do this proof another day.
5
2
4
1
3
6
Example # 1.
If a convex polygon has 24 sides (24-gon), then …
3960º
(a) … the interior angle sum is ___________.
(n - 2)  180º = (24 - 2)  180º
= (22)  180º
= 3960º
360º
(b) … the exterior angle sum is ___________.
Example # 2. (a)
Find the value of “x”.
x + 150 + 90 + 90 + 160 = 540
160º
xº
x + 490 = 540
x = 50
150º
n=5
angle sum = 540º
50
x = _____
Example # 2. (b)
Find the value of “x”.
x + 60 + 150 + 50 = 360
xº
x + 260 = 360
60º
50º
x = 100
150º
n=4
angle sum = 360º
x = _____
100
Example # 2. (c)
Find the value of “x”.
x + 140 + 120 + 60 + 160 + 150 = 720
140º
x + 630 = 720
120º
60º
xº
150º
x = 90
160º
90
x = _____
n=6
angle sum = 720º
Example # 2. (d)
Find the value of “x”.
60º
xº
xº
xº
n=4
angle sum = 360º
x + x + x + 60 = 360
3 x = 300
x = 100
100
x = _____
REGULAR POLYGON:
A polygon that is both
equiangular and equilateral .
template #3
template #2
template #4
Look at some examples using your template.
REGULAR POLYGON
The angle measure of a regular polygon with n sides …
… has every interior angle =
… has every exterior angle =
n = the number of sides,
which is also the number of angles.
(n - 2) ×180°
n
360°
n
Example # 3.
Find the measure of each interior angle and each
exterior angle of a regular pentagon.
108º
Each interior angle has measure _____.
(n - 2) ×180° (5 - 2) ×180° (3) ×180°
=
= 108°
=
5 These angles
n
5
will always be
Each exterior angle has measure _____.
72º supplementary!
360° 360°
=
5
n
= 72°
Example # 4.
How many sides does a regular polygon have
if the measure of each exterior angle is 45º?
8 sides.
The polygon has ___
360°
= 45°
n
360°
=n
45°
n=8
Example # 5.
How many sides does a regular polygon have
if the measure of each interior angle is 150º?
The polygon has ___
12 sides.
(n - 2)180°
= 150°
n
180º n - 360º = 150º n
30º n = 360º
n = 12
Example # 5. A better way!
How many sides does a regular polygon have
if the measure of each interior angle is 150º?
The polygon has ___
12 sides.
Remember:
interior angle + exterior angle = 180º
150º + exterior angle = 180º
exterior angle = 30º
360°
= 30°
n
360°
=n
30°
n = 12
Assignment
Written Exercises on pages 104 & 105
RECOMMENDED: 11, 13, 15, 19
REQUIRED: 8, 10, 16, 17, 20, 21, 22, 23
Prepare for a quiz on Lessons 3-4 & 3-5
How can you apply parallel
lines (planes) to make
deductions?