Download Angles Intro , tranversals

Introduction
to
angles
Introduction to Angles and
Triangles
Math is a language
Line – extends indefinitely, no thickness
or width
Ray – part of a line, starts at a point, goes
indefinitely
Line segment – part of a line, begin and
end point
Angle - two lines, segments
or rays from a common point
Vertex - common point at which two lines
or rays are joined
Degrees: Measuring Angles
We measure the size of an angle using degrees.
Example: Here are some examples of angles and
their degree measurements.
Acute Angles
An acute angle is an angle measuring
between 0 and 90 degrees.
Example:
Right Angles
A right angle is an angle measuring
90 degrees.
Example:
90°
Complementary Angles
Two angles are called complementary angles if the
sum of their degree measurements equals 90 degrees.
Example:
These two angles are complementary.
32°
58°
Together they create a 90° angle
Obtuse Angles
An obtuse angle is an angle measuring
between 90 and 180 degrees.
Example:
Straight Angle
A right angle is an angle measuring 180 degrees.
Examples:
Supplementary Angles
Two angles are called supplementary angles if the sum
of their degree measurements equals 180 degrees.
Example:
These two angles are supplementary.
139°
41°
These two angles sum is 180° and together
the form a straight line
Review
State whether the following are acute, right, or obtuse.
acute
1.
2.
?
acute
3.
4.
5.
right
?
obtuse
obtuse
Complementary and Supplementary
Find the missing angle.
1. Two angles are complementary. One measures 65 degrees.
Answer : 25
2. Two angles are supplementary. One measures 140 degrees.
Answer : 40
Complementary and Supplementary
Find the missing angle. You do not have a protractor.
Use the clues in the pictures.
1.
2.
x
55
X=35
x
165
X=15
1.
x 90
90
y z
90
x = 90
y = 90
z = 90
2.
70x 110 
y z
110
x = 70
y = 110
z = 70
Vertical Angles are angles on opposite sides of
intersecting lines
1.
90 and y are vertical angles
x 90
90
90
y 90
z
x and z are vertical angles
The vertical angles in this case are
equal, will this always be true?
110 and y are vertical angles
2.
x70 110
110
y 70
z
x and z are vertical angles
Vertical angles are always equal
Vertical Angles
Find the missing angle.
Use the clues in the pictures.
58
x
X=58
Can you find the missing angles?
F
E
D
J
20
90
70
C
70
90
20
H
G
Can you find these missing angles
B
A
C
52
60
F
68
G
68
60
D
52
E
Parallel lines
transversals
and their
angles
18
Parallel Lines
You will learn to identify the relationships among pairs of
interior and exterior angles formed by two parallel lines
and a transversal.
In geometry, two lines in a plane that never intersect ,
parallel lines
have the same slope, are called ____________.
parallel lines are always the same distance apart
Parallel Lines and Transversals
In geometry, a line, line segment, or ray that intersects two or more lines at
transversal
different points is called a __________
The lines cut by a transversal may or may not be parallel.
Parallel Lines
Nonparallel Lines
l
1 2
4 3
lm
1 2
4 3
m
5 6
8 7
t
t is a transversal for l and m.
b
c
5 6
8 7
b || c
r
r is a transversal for b and c.
Parallel Lines and Transversals
AB is an example of a transversal. It intercepts lines l and m.
• We will be most concerned
with transversals that cut
parallel lines.
• When a transversal cuts
parallel lines, special pairs of
angles are formed that are
sometimes congruent and
sometimes supplementary.
A
2
1
4
5
8
B
6
7
3
l
m
Parallel Lines and Transversals
Two lines divide the plane into three regions.
The region between the lines is referred to as the interior.
The two regions not between the lines is referred to as the exterior.
Exterior
Interior
Exterior
Parallel Lines and Transversals
eight angles are formed.
When a transversal intersects two lines, _____
These angles are given special names.
1. Interior angles , 3,4,5,6
lie between the two parallel lines.
2. Exterior angles 1,2,7,8
lie outside the two lines.
3. Alternate Interior angles 4&6, 5&3
opposite sides of the transversal and
lie between the parallel lines
l
1 2
4 3
m
5 6
8 7
t
5. Consecutive Interior angles 4&5, 3&6
4. Alternate Exterior angles 1&7, 2&8 on the same side of the transversal
and are between the parallel lines
Are on the opposite sides of the
transversal and lie outside the
6. Corresponding angles
two lines
1&5, 4&8, 2&6, 3&7
on the same side of the transversal
one is exterior and the other is
interior
Name the pairs of the following angles formed by a
transversal.
GG
G
AA
500
Line
Line
Line LL
L
P
P
BBB
1300
D
DD
Q
Q
Q
EEE
FFF
Line
Line
Line
MM M
Line
Line
N
Line
NN
Congruent:
Same shape and size
The symbol 
means that the shapes, lines
or angles are congruent
two shapes both have an area of 36 in2 , are they congruent?
9 in
6 in
4 in
6 in
Area is 36
in2
Area is 36 in2
Numbers, or expressions can have equal value…..
In Geometry, we use “congruent” to describe two or
more objects, lines or angles as being the same
Parallel Lines and Transversals
Alternate interior angles are congruent
_________.
4  6
3  5
1 2
4 3
5 6
8 7
Parallel Lines and Transversals
congruent
Alternate exterior angles is _________.
1  7
2  8
1 2
4 3
5 6
8 7
Parallel Lines and Transversals
supplementary
consecutive interior angles is _____________.
4  5  180
3  6  180
1 2
4 3
5 6
8 7
Transversals and Corresponding Angles
congruent
corresponding angles is _________.
Transversals and Corresponding Angles
Types of angle pairs formed when
a transversal cuts two parallel lines.
Concept
Summary
Congruent
Supplementary
alternate interior
consecutive interior
alternate exterior
corresponding
Transversals and Corresponding Angles
s
s || t and c || d.
Name all the angles that are
congruent to 1.
Give a reason for each answer.
1 2
5 6
9
10
13 14
3  1
corresponding angles
6  1
vertical angles
8  1
alternate exterior angles
9  1
corresponding angles
14  1
alternate exterior angles
11  9  1
corresponding angles
16  14  1
corresponding angles
t
3
7
11 12
15 16
c
4
8
d
Let’s Practice
120° 1
60°
3
120° 5
60°
7
2 60°
4 120°
6 60°
8
120°
m<1=120°
Find all the remaining angle
measures.
Another practice problem
40°
60° 6
80°
100°
80°
11 9
12
80°10 100°
8
5
40°
7
4
180-(40+60)= 80°
60°
120°
1
3
120°
60°
60° 2
Find all the missing
angle measures,
and name the
postulate or
theorem that gives
us permission to
make our
statements.