Download Geometry Section 3.3 Notes

CHAPTER
3
3.3 Polygons and
Angles
Copyright © 2014 Pearson Education, Inc.
Slide 6-1
Definitions
Polygon Definition
A figure is a polygon if it meets the following
conditions:
1. It is a plane figure formed by three or more line
segments called sides.
2. Sides that have a common endpoint are
noncollinear.
3. Each side intersects exactly two other sides, but
only at their endpoints.
Copyright © 2014 Pearson Education, Inc.
Slide 6-2
Polygons
The endpoints of the sides of a polygon are called the
vertices (singular is vertex).
Below are some examples of polygons. Each vertex of the
middle polygon is labeled. A polygon can be named by
listing its vertices consecutively in order, as shown.
Polygon DEFGH 
 two of many correct names
Polygon EDHGF 
Copyright © 2014 Pearson Education, Inc.
Slide 6-3
Names of Polygons
Number of Sides
3
Name of Polygon
triangle
4
5
6
7
quadrilateral
pentagon
hexagon
heptagon
8
9
10
octagon
nonagon
decagon
12
n
dodecagon
n-gon
Copyright © 2014 Pearson Education, Inc.
Slide 6-4
Example
Identifying Polygons
Identify the polygons. If not a polygon, state why.
a.
b.
c.
d.
e.
Solution
Figures A and E are polygons.
Figure B is not a polygon because there is a “side”
that is a curve and not a line segment.
Copyright © 2014 Pearson Education, Inc.
Slide 6-5
Example
Identifying Polygons
Identify the polygons. If not a polygon, state why.
a.
b.
c.
d.
e.
Solution
Figure C is not a polygon by our definition because
there is a side that intersects more than two other
sides.
Figure D is not a polygon because two sides
intersect only one other side.
Copyright © 2014 Pearson Education, Inc.
Slide 6-6
Definitions
In general, a polygon with n sides is called an ngon. For example, a polygon with 13 sides is called
a 13-gon.
Another way to classify polygons is as convex or
concave. A polygon is convex if no line containing
a side contains a point within the interior of the
polygon. A polygon is concave (or nonconvex) if it
is not convex.
Copyright © 2014 Pearson Education, Inc.
Slide 6-7
Identifying Convex and Concave
Example
Polygons
Identify the polygons. If not a polygon, state why.
a.
b.
c.
Solution
a. The polygon has 8 sides, so it is an octagon. None
of the extended sides contain a point of the interior,
so it is convex.
Copyright © 2014 Pearson Education, Inc.
Slide 6-8
Identifying Convex and Concave
Example
Polygons
Identify the polygons. If not a polygon, state why.
a.
b.
c.
Solution
b. The polygon has 6 sides, so it is a hexagon. Some
of the extended sides contain a point of the interior,
so it is concave (or nonconvex).
Copyright © 2014 Pearson Education, Inc.
Slide 6-9
Identifying Convex and Concave
Example
Polygons
Identify the polygons. If not a polygon, state why.
a.
b.
c.
Solution
c. The polygon has 4 sides, so it is a quadrilateral.
None of the extended sides contain a point within
the interior, so it is convex.
Copyright © 2014 Pearson Education, Inc.
Slide 6-10
Definition
An equilateral polygon is a
polygon with all sides congruent.
An equiangular polygon is a
polygon with all angles congruent.
A regular polygon is a polygon
that is both equilateral and equiangular.
Copyright © 2014 Pearson Education, Inc.
Slide 6-11
Example
Identifying Regular Polygons
Determine if each polygon is regular or not. Explain
your reasoning.
a.
b.
c.
Solution
a. The pentagon is equilateral and equiangular, so it
is a regular polygon.
Copyright © 2014 Pearson Education, Inc.
Slide 6-12
Example
Identifying Regular Polygons
Determine if each polygon is regular or not. Explain
your reasoning.
a.
b.
c.
Solution
b. The hexagon is equilateral, but not equiangular,
so it is not a regular polygon.
Copyright © 2014 Pearson Education, Inc.
Slide 6-13
Example
Identifying Regular Polygons
Determine if each polygon is regular or not. Explain
your reasoning.
a.
b.
c.
Solution
c. The quadrilateral is equilateral, but not
equiangular, so it is not a regular polygon.
Copyright © 2014 Pearson Education, Inc.
Slide 6-14
Definitions
The perimeter P of a polygon is the sum of the
lengths of its sides.
Copyright © 2014 Pearson Education, Inc.
Slide 6-15
Finding the Perimeter of an
Example
Irregular Room
Find the perimeter of the room.
Solution
To find the perimeter of the room,
we first need to find the lengths of
all sides of the room.
Copyright © 2014 Pearson Education, Inc.
Slide 6-16
Finding the Perimeter of an
Example
Irregular Room
Add the measures to find the perimeter.
perimeter = 10 ft + 9 ft + 3 ft + 6 ft + 7 ft + 15 ft
= 50 ft
The perimeter of the room is 50 feet.
Copyright © 2014 Pearson Education, Inc.
Slide 6-17
Triangle Interior Angle Sum Worksheet
Copyright © 2014 Pearson Education, Inc.
Slide 6-18
Corollary 3.14 Exterior Angle of a Triangle
The measure of each exterior angle of a triangle
equals the sum of the measures of its two
nonadjacent interior angles.
m∠1 = m∠2 + m∠3
Copyright © 2014 Pearson Education, Inc.
Slide 6-19
Corollary 3.13 Third Angles Theorem
Copyright © 2014 Pearson Education, Inc.
Slide 6-20
Example
Using the Third Angles Theorem
Find the value of x.
Solution
From the figures, we have
∠J ≅ ∠R and ∠H ≅ ∠S.
Thus, from the Third Angles Theorem,
∠K ≅ ∠Q or m∠K = m∠Q.
Use ΔJHK to find m∠K.
m∠K = 180° – 53° – 92° = 35°
Copyright © 2014 Pearson Education, Inc.
Slide 6-21
Example
Using the Third Angles Theorem
Find the value of x.
Solution
m∠K = m∠Q
35 = 10x + 5
30 = 10x
3=x
The value of x is 3. To check, replace x with 3 and
see that 10x + 5 = 35. Then make sure that
53° + 92° + 35° = 180°.
Copyright © 2014 Pearson Education, Inc.
Slide 6-22
Example
Finding Angle Measures
Use the Triangle Angle-Sum Theorem to find the
measure of each angle in the given triangle.
Solution
5x + 6x + 15x + 24 = 180
26x + 24 = 180
26x = 156
x=6
Now let’s use the value of x and the given triangle to
find the measure of each angle.
Copyright © 2014 Pearson Education, Inc.
Slide 6-23
Example
Finding Angle Measures
Use the Triangle Angle-Sum Theorem to find the
measure of each angle in the given triangle.
Solution
If x = 6,
then 5x = 5(6) = 30
6x = 6(6) = 36
15x + 24 = 15(6) + 24 = 90 + 24 = 114
Check: 30° + 36° + 114° = 180°
Copyright © 2014 Pearson Education, Inc.
Slide 6-24
Example
Finding Angle Measures
Use the Exterior Angle of a Triangle Corollary to
find the measure of the exterior angle and the
nonadjacent angle shown.
Solution
3x – 53 = x + 67
2x – 3 = 67
2x = 120
x = 60
Copyright © 2014 Pearson Education, Inc.
Slide 6-25
Example
Finding Angle Measures
Solution
Let’s use the value of x
and the given figure to find
the measure of each angle.
x = 60
3x – 53 = 3(60) – 53 = 180 – 53 = 127
Also, x° = 60°
Copyright © 2014 Pearson Education, Inc.
Slide 6-26
Definition
A segment joining two nonconsecutive vertices of a
convex polygon is called a diagonal of the polygon.
Copyright © 2014 Pearson Education, Inc.
Slide 6-27
Interior Angle Sum Worksheet
Copyright © 2014 Pearson Education, Inc.
Slide 6-28
Theorem 3.15 Polygon Interior Angle-Sum
Theorem
The sum of the measures of the interior angles of a
convex n-gon is (n – 2) * 180°
Copyright © 2014 Pearson Education, Inc.
Slide 6-29
Corollary 3.16 Regular Polygon Interior
Angle Corollary (Theorem to 10.1-1)
The measure of each interior angle of a regular
n-gon is
1
(n  2) 180
 (n  2) 180 , or
n
n
Copyright © 2014 Pearson Education, Inc.
Slide 6-30
Finding the Sum of the Measures
Example
of the Angles of a Polygon
Find the sum of the measures of the interior angles
of a convex octagon.
Solution
An octagon has 8 sides.
 (n  2) 180
 (8  2) 180
 6 180
 1080
The interior angle sum of a convex octagon is
1080°.
Copyright © 2014 Pearson Education, Inc.
Slide 6-31
Finding the Measure of an Interior
Example
Angle
Find the value of x in the figure. Then use x to find
mA and mB.
Solution
This is a hexagon, which has 6 sides.
The sum of the interior angles of any convex
hexagon is:
sum of angles = (6 – 2) 180°
= 4 180° = 720°
Copyright © 2014 Pearson Education, Inc.
Slide 6-32
Finding the Measure of an Interior
Example
Angle
To find x, let’s add the interior angle measures of
the polygon and set the sum equal to 720°.
mA  mB  mC  mD  mE  mF  720
( x  20)  x  135  90  125  90  720
2 x  460  720
2 x  260
x  130
mB  x  130
mA  ( x  20)  (130  20)  150
Copyright © 2014 Pearson Education, Inc.
Slide 6-33
Using the Regular Polygon
Example
Interior-Angle Corollary
The Sino-Steel Tower is a hexagonal, honey comb-looking
“green” building, in Tianjin, China, designed by MAD
Studios architects. Find the measure of each interior angle
of one regular hexagon.
Solution each angle  (n  2) 180
n
(6  2) 180

6
4 180

6
 120
Copyright © 2014 Pearson Education, Inc.
Slide 6-34
Finding the Number of Sides of a
Example
Regular Polygon
The measure of an interior angle of a regular
polygon is 144°. Find the number of sides of this
polygon.
Solution
36n  360
(n  2) 180
 144
n  10
n
(n  2) 180
n
 n 144
The regular polygon
n
(n  2) 180  144n
has 10 sides.
180n  360  144n
Copyright © 2014 Pearson Education, Inc.
Slide 6-35
Example
Solve for x
Find x, and then the measures of angles C and D.
Solution
x + (2x + 3) + 127 + 125 = 360
3x + 255 = 360
3x = 105
x = 35
Copyright © 2014 Pearson Education, Inc.
Slide 6-36
Example
Solve for x
Find x, and then the measures of angles C and D.
Solution
Use x = 35 to find m∠C and m∠D.
m∠D = x° = 35°
m∠C = (2x + 3)° = 2(35) + 32° = 73°
Copyright © 2014 Pearson Education, Inc.
Slide 6-37
Exterior Angles
The angles that are adjacent to the interior angles of
a convex polygon are the exterior angles of the
polygon.
Exterior angles are 4, 5,
6, 7, 8 and 9
Copyright © 2014 Pearson Education, Inc.
Slide 6-38
Theorem 3.17 Polygon Exterior Angle-Sum
Theorem
Copyright © 2014 Pearson Education, Inc.
Slide 6-39
Corollary 3.18 Regular Polygon Exterior
Angle Corollary (to Theorem 3.17)
Copyright © 2014 Pearson Education, Inc.
Slide 6-40
Finding the Number of Sides of a
Example
Regular Polygon
Find the measure of each exterior angle of a regular
pentagon.
Solution
Since this is a regular polygon,
each exterior angle has the same
measure.
360
m1  m2  m3  m4  m5 
 72
5
Each exterior angle measures 72 degrees.
Copyright © 2014 Pearson Education, Inc.
Slide 6-41
Finding the Number of Sides of a
Example
Regular Polygon
Find the value of x. Then find
each exterior angle measure.
Solution
The sum of the exterior
angles is 360°
x  2 x  2 x  (4 x  7)  (5 x  3)  360
14 x  10  360
14 x  350
x  25
Copyright © 2014 Pearson Education, Inc.
Slide 6-42
Finding the Number of Sides of a
Example
Regular Polygon
The angle measures are:
x  25
2x  2  25  50
4 x  7  4  25  7  107
5 x  3  5  25  3  128
Copyright © 2014 Pearson Education, Inc.
Slide 6-43