Download Angle Pairs

7-1 7-2
Angles
PA
Measurement of an Angle
2
To denote the measure of an angle we write an
“m” in front of the symbol for the angle.
Here are some common angles and their measurements.
m1  45
1
m2  90
2
m3  135
3
4
m4  180
3
Congruent Angles
• So, two angles are congruent if and only if
they have the same measure.
• So,
The angles below are congruent.
Means
Congruent
Means Equal
A  B if and only if mA  mB.
4
Types of Angles
• An acute angle is an angle that measures less than
90 degrees.
• A right angle is an angle that measures exactly 90
degrees.
• An obtuse angle is an angle that measures more
than 90 degrees.
acute
right
obtuse
5
Types of Angles
• A straight angle is an angle that measures 180
degrees. (It is the same as a line.)
• When drawing a right angle we often mark its
opening as in the picture below.
right angle
straight angle
6
Perpendicular Lines
• Two lines are perpendicular if
they intersect to form a right
angle. See the diagram.
• Suppose angle 2 is the right
angle. Then since angles 1 and 2
are supplementary, angle 1 is a
right angle too. Similarly, angles
3 and 4 are right angles.
• So, perpendicular lines intersect
to form four right angles.
2
1
3 4
7
Perpendicular Lines
• The symbol for perpendicularity is
.
• So, if lines m and n are perpendicular, then we write
m  n.
m
mn
n
Adjacent Angles
Adjacent angles share a common vertex and
one common side.
Adjacent angles are “side by side”
and share a common ray.
15º
45º
Adjacent Angles
These are examples of adjacent angles.
80º
45º
35º
55º
130º
85º
20º
50º
Adjacent Angles
These angles are NOT adjacent.
100º
50º
35º
35º
55º
45º
Vertical Angles
• Two angles formed by intersecting lines and have no
sides in common but share a common vertex.
• Are congruent.
Common
Vertex
75º
When 2 lines
intersect, they
make vertical
angles.
105º
105º
75º
Vertical Angles
Vertical
angles are
opposite
one
another.
75º
105º
105º
75º
Vertical Angles
Vertical
angles are
opposite
one
another.
75º
105º
105º
75º
Vertical Angles
Vertical angles are congruent
(equal).
150º
30º
30º
150º
Vertical Angles
Two angles that are opposite angles.
Vertical angles are congruent.
Name the Vertical Angles
14
 5   8,
1
23
3
5
7
4
6
8
2
67
Supplementary Angles
Add up to 180º.
40º
120º
60º
Adjacent and
Supplementary Angles
140º
Supplementary Angles
• Two angles are supplementary if their
measures add up to 180.
• If two angles are supplementary each angle is
the supplement of the other.
• If two adjacent angles together form a straight
angle as below, then they are supplementary.
1
2
1 and 2 are
supplementary
18
Complementary Angles
Add up to 90º.
20º
20º
70º
Adjacent and
Complementary Angles
70º
Complementary Angles
• Two angles are if their measures add
upcomplementary to 90.
• If two angles are complementary, then each
angle is called the complement of the other.
• If two adjacent angles together form a right
angle as below, then they are complementary.
1 and 2 are
complementary
if ABC is a
A
1
B
2 C
right angle
20
Supplementary vs. Complementary
How do I remember?
The way I remember is this:
• C comes before S in the alphabet.
• 90 comes before 180 when I count.
• Complementary is 90, Supplementary is 180.
Guess Who?
• I am an angle.
Guess Who?
• I am an angle.
• I have 180°
Guess Who?
• I am an angle.
• I have 180°
• I look like this:
Guess Who?
• I am an angle.
• I have 180°
• I look like this:
Supplementary
Complementary
Guess Who?
• I am two adjacent angles.
Guess Who?
• I am two adjacent angles.
• I look like an “L” with a line in the middle.
Guess Who?
•
•
•
•
I am two adjacent angles.
I look like an “L” with a line in the middle.
I add up to 90°
I look like this:
Guess Who?
•
•
•
•
I am two adjacent angles.
I look like an “L” with a line in the middle.
I add up to 90°
I look like this:
Complementary
Supplementary
Guess Who?
Complementary
Supplementary
Guess Who?
Complementary
Supplementary
Review
• Complementary angles
are…….
Review
• Complementary angles
are…….
Review
• Supplementary Angles
are…..
Practice Time!
Find the missing angle
I know that these angles are
complementary.
They must add up to 90°
So……
90 – 55 = 35
The missing angle is 35
55
x
You try.
Are they
supplementary or
complementary?
x
20
Find the missing
side.
You try.
Are they
supplementary or
complementary?
complementary
Find the missing
side.
x
20
90 – 20 = 70
The missing angle
Is 70
One More
50
x
Find the missing angle
120
I know these are supplementary angles.
Supplementary angles add up to 180.
The given angle is 120. So…..
180 – 120 = 60
The missing angle is 60
x
Find the missing angle
130
What kind of angles?
What’s the missing angle?
x
Find the missing angle
130
What kind of angles?
Supplementary
What’s the missing angle?
Adds up to 180, so…..
180 – 130 = 50
x
Find the missing angle
x
Do this one on your own.
30
Directions:
Identify each pair of angles as
adjacent, vertical, supplementary,
complementary,
or none of the above.
#1
120º
60º
#1
120º
60º
Supplementary Angles
Adjacent Angles
#2
30º
60º
#2
30º
60º
Complementary Angles
#3
75º
75º
#3
Vertical Angles
75º
75º
#4
40º
60º
#4
40º
60º
None of the above
#5
60º
60º
#5
60º
60º
Vertical Angles
#6
135º
45º
#6
135º
45º
Supplementary Angles
Adjacent Angles
#7
25º
65º
#7
25º
65º
Complementary Angles
Adjacent Angles
#8
90º
50º
#8
90º
50º
None of the above
Directions:
Determine the missing angle.
#1
?º
45º
#1
135º
45º
#2
?º
65º
#2
25º
65º
#3
?º
35º
#3
35º
35º
#4
?º
50º
#4
130º
50º
#5
?º
140º
#5
140º
140º
#6
?º
40º
#6
50º
40º
Transversal
• Definition: A line that intersects two or more lines in a
plane at different points is called a transversal.
• When a transversal t intersects line n and m, eight angles
of the following types are formed:
Exterior angles
Interior angles
Consecutive interior angles
Alternative exterior angles
Alternative interior angles
Corresponding angles
m
n
t
Corresponding Angles
Corresponding Angles: Two angles that occupy
corresponding positions.
The corresponding angles are the ones at the same location
at each intersection
26
3
7
15
1
3
5
7
4
8
2
4
6
8
75
Angles and Parallel Lines
•
1.
2.
3.
If two parallel lines are cut by a transversal, then the
following pairs of angles are congruent.
Corresponding angles
Alternate interior angles
Alternate exterior angles
Proving Lines Parallel
• If two lines are cut by a transversal and corresponding
angles are congruent, then the lines are parallel.
A
B
C
D
Alternate Angles
• Alternate Interior Angles: Two angles that lie between parallel
lines on opposite sides of the transversal (but not a linear pair).
 3   6,  4   5
• Alternate Exterior Angles: Two angles that lie outside parallel
lines on opposite sides of the transversal.  2   7,  1   8
1
3
5
7
2
4
6
8
Lesson 2-4: Angles and Parallel Lines
78
Example: If line AB is parallel to line CD and s is parallel to t, find
the measure of all the angles when m< 1 = 100°. Justify your answers.
A
1
4
C
5
8
m<2=80° m<3=100° m<4=80°
2
12
3
6
10
11
B
D
13 14
16 15
7
s
9
t
m<5=100° m<6=80° m<7=100° m<8=80°
m<9=100° m<10=80° m<11=100° m<12=80°
m<13=100° m<14=80° m<15=100° m<16=80°
Lesson 2-4: Angles and Parallel Lines
79
Proving Lines Parallel
• If two lines are cut by a transversal and alternate
interior angles are congruent, then the lines are parallel.
A
B
C
D
Ways to Prove Two Lines Parallel
• Show that corresponding angles are equal.
• Show that alternative interior angles are equal.
• In a plane, show that the lines are perpendicular to the
same line.
Homework
Pg 305 #6-14e, 18-32e (just answers)
Pg 309 #6-24e (just answers)