Transcript polygon - Cloudfront.net

1.5: Describe Angle Pair Relationships
1.6: Classify Polygons
Objectives:
1. To use special angle relationships to find
angle measures
2. To define, name, and classify polygons
Vocabulary (make sure you know these)
Complementary
Supplementary
Linear Pair
Vertical Angles
Polygon
Diagonal (n.)
Convex
Concave
Equilateral
Equiangular
Regular
Define these in
your notebook
C Comes Before S…
m1  m2  90
m3  m4  90
m5  m6  180
m7  m8  180
Example 1a
1. Given that <1 is a complement of <2 and
m<1 = 68°, find m<2.
220
2. Given that <3 is a supplement of <4 and
m<3 = 56°, find m<4.
1240
Example 1b
1. What is the sum of complementary angles
in radians? Π
2
2. What is the sum of supplementary angles
in radians? Π
3. What is complement for the angle that
measures π/3? Π6
4. What is the supplement for the angle that
measures 3π/4? Π
4
Example 2
Let <A and <B be complementary angles
and let m<A = (2x2 + 35)° and m<B =
(x + 10)°. What is (are) the value(s) of x?
What are the measures of the angles?
Set up the equation and solve
X = 4.5 or -5
m<A = 14.5 or 85
m<B = 75.5 or 5
Check to make sure the
sum is 90
Linear Pairs of Angles
Linear Pairs of Angles
• Two adjacent angles
form a linear pair if
their noncommon
sides are opposite
rays.
• The angles in a linear
pair are
supplementary.
Vertical Angles
Vertical Angles
• Two nonadjacent
angles are vertical
angles if their sides
form two pairs of
opposite rays.
• Vertical angles are
formed by two
intersecting lines.
Check them out HERE
Example 3
Identify all of the linear pairs of angles and all
of the vertical angles in the figure.
Example 4: SAT
y
z
In the figure  5 and  4 , what is the value
x
x
of x?
x=18, y=90 and z=72
HOW did I do that?
x
y
z
3-D Rendering
3-D rendering in digital graphics is based
upon polygons.
3-D Rendering
The higher the
polygon count, the
smoother the
surface.
– Tomb Raider
(1996)
3-D Rendering
The higher the
polygon count, the
smoother the
surface.
– Tomb Raider
Underworld (2008)
What Makes a Polygon?
So, what makes a polygon a polygon?
Polygons
A closed plane figure is
a polygon if it is
formed by 3 or more
line segments (sides),
joined endpoint to
endpoint (vertices)
with each side
intersecting exactly
two others.
Parts of a Polygon
Consecutive
Angles
Consecutive
Vertices
Consecutive Sides
What’s the name of this polygon?
Example 5
Why are the following not polygons?
Names of Polygons (memorize these)
Sides
Name
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
11
Undecagon
12
Dodecagon
• Polygons come in many
flavors.
• They are classified by
the number of sides they
have.
• A polygon with more
than 12 sides is
commonly called an
n-gon, where n is the
number of sides.
Names of Polygons
Sides
Name
Sides
Name
Triangle
13
Tridecagon
4
Quadrilateral*
14
Tetradecagon
5
Pentagon
15
Pentadecagon
6
Hexagon
16
Hexadecagon
7
Heptagon
17
Heptadecaton
8
Octagon
18
Octadecagon
9
Nonagon
19
Nonadecagon
10
Decagon
20
Icosagon
11
Undecagon
100
Hectagon
12
Dodecagon
*Also called a Tetragon
3
1,000,000 Hecatommyriagon
Example 6
Name each polygon.
hexagon
C
B
quadrilateral
T
W
A
F
D
C
G
U
Example 7
When you buy a
42” television,
how or where is
that 42 inches
measured?
Diagonal
Diagonal
A diagonal is a line
segment that joins two
nonconsecutive
vertices of a polygon.
Example 8
How many diagonals are there in an
octagon? (Do you really want to draw that?
Heck no! In your notebook make a table
and find a pattern!)
Convex & Concave Polygons
Convex & Concave Polygons
Convex polygons have
all their diagonals in
the interior of the
polygon.
Concave polygons
have at least one
diagonal on the
exterior of the
polygon.
Example 9
Tell whether the figure is a polygon and
whether it is convex or concave.
Equilateral Polygon
An equilateral polygon
is a polygon in which
all of its sides are
congruent.
Equiangular Polygon
An equiangular
polygon is a polygon
in which all its interior
angles are congruent.
Regular Polygon
A regular polygon is a
polygon that is
equilateral and
equiangular.
Example 10
Classify the polygon by the number of sides.
Tell whether the polygon is equilateral,
equiangular, or regular. Explain your
reasoning.
Example 11: SAT
In the figure, RS = ST
and the coordinates
of S are (k, 3).
What is the value of
k?
y
S
T
x
-3
R
O
(1, 0)
Example 12
Given that the figure
is regular, find the
values of x and y.
x=12, y=8