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## Angles and Parallel Lines

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Title: Angles and Parallel Lines

1
Angles and Parallel Lines
• Geometry D Section 3.2

2
Angles and Parallel Lines
We are going to investigate the relationship of
various angles created by two parallel lines and
a transversal.
Obtain a ½ sheet of graph paper and a protractor.
Construct two lines and a transversal similar
to the image on the next slide.
3
Angles and Parallel Lines
Extend your lines the full height and width of
the paper.
Pause for time to work!
4
Angles and Parallel Lines
Label the angles as shown below.
1
2
3
4
6
5
7
8
Pause for time to work!
5
Angles and Parallel Lines
Measure all angles using a protractor to the
nearest degree.
1
2
3
4
6
5
7
8
Pause for time to work!
6
Angles and Parallel Lines
Measure all angles using a protractor to the
nearest degree.
127o
53o
1
2
Note Your measurements may be different values
but should be in the same pattern.
3
4
53o
127o
127o
53o
6
5
7
8
53o
127o
7
Angles and Parallel Lines
Identify the relationship between the following
angles?
127o
53o
1
2
3
4
53o
127o
127o
53o
6
5
7
8
53o
From Chapter 2, the angles are linear pairs.
127o
What can be said about the measures of the linear
pairs?
Linear pairs are supplementary (sum to 180o).
8
Angles and Parallel Lines
Identify the relationship between the following
angles?
127o
53o
1
2
3
4
53o
127o
127o
53o
6
5
7
8
53o
From Chapter 2, the angles are vertical angles.
127o
What can be said about the measures of the
vertical angles?
Vertical angles are congruent angles.
9
Angles and Parallel Lines
Identify the relationship between the following
angles?
127o
53o
1
2
3
4
53o
127o
127o
53o
6
5
7
8
53o
The angles are corresponding angles.
127o
What can be said about the measures of the
corresponding angles?
The measures are equal and the angles are
congruent.
10
Angles and Parallel Lines
Corresponding Angles Postulate
If two parallel lines are cut by a transversal,
then each pair of corresponding angles are
congruent.
11
Angles and Parallel Lines
Identify the relationship between the following
angles?
127o
53o
1
2
3
4
53o
127o
127o
53o
6
5
7
8
53o
The angles are alternate interior angles.
127o
What can be said about the measures of the
alternate interior angles?
The measures are equal and the angles are
congruent.
12
Angles and Parallel Lines
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal,
then each pair of alternate interior angles are
congruent.
You will prove this theorem as a homework problem!
13
Angles and Parallel Lines
Identify the relationship between the following
angles?
127o
53o
1
2
3
4
53o
127o
127o
53o
6
5
7
8
53o
The angles are alternate interior angles.
127o
What can be said about the measures of the
alternate interior angles?
14
Angles and Parallel Lines
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal,
then each pair of consecutive interior angles is
supplementary (sum to 180o).
You will prove this theorem as a homework problem!
15
Angles and Parallel Lines
Identify the relationship between the following
angles?
127o
53o
1
2
3
4
53o
127o
127o
53o
6
5
7
8
53o
The angles are alternate exterior angles.
127o
What can be said about the measures of the
alternate interior angles?
The measures are equal and the angles are
congruent.
16
Angles and Parallel Lines
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal,
then each pair of alternate exterior angles is
congruent.
Prove
Statement Reason
?
p q, t is a transversal of p q
Given
t
Corresponding s are
?
1
2
p
3
4
?
6
5
Vertical s are
q
7
8
?
Transitive Property
17
Angles and Parallel Lines
Perpendicular Transversal Theorem
In a plane, if a line is perpendicular to one of
two perpendicular lines, then it is perpendicular
to the other.
t
p
q
If t is perpendicular ( ) to p, then it is
also perpendicular to q.
You will prove this theorem as a homework problem!
18
Angles and Parallel Lines
Applications
Gather into groups of not more than 3. Work the
following problems in your group. Compare your
19
Angles and Parallel Lines
Given j k,
Applications
Make a sketch of the problem in your notes.
Find the measure of
4
5
3
1.
43o
2
1
7
8
6
Corresponds with 1.
9
11
10
2.
24o
12
Alternate exterior with 14.
13
3.
156o
Linear pair with 9. 180o 24o 156o
14
20
Angles and Parallel Lines
Given j k, Find the measure of
Applications
4
5
3
4.
137o
2
1
7
8
6
Linear pair with 3.
9
11
10
5.
156o
12
Vertical angle with 10.
13
6.
43o
Vertical with 1. Alternate Interior of
3.
14
21
Angles and Parallel Lines
Applications
Find the values of x and y in each figure. Find
the measure of each given angle. Note Figures
are not drawn to scale.
Given
Pause for time to work!
22
Angles and Parallel Lines
Applications Solution
Given
Linear pairs are supplementary.
(5x 2) (9x 10) 180o
14x 12 180
14x 168
x 12
Linear Pair
By corresponding angles,
62o
118o
3y 1 62
62o
3y 63
y 21 and
23
Angles and Parallel Lines
Applications
Find the values of x, y and z in each figure.
(2z)o
(3x3)o
(4y2)o
66o
Pause for time to work!
24
Angles and Parallel Lines
is a corresponding angle with the
angle of 66o.
Applications
(3x 3)o and 66o are linear pairs and sum to
180o.
(3x 3)o 66o 180o 3x 63 180 3x 117,
x 39
(2z)o
(3x3)o
66o
(4y 2)o and 66o are congruent alternate
interior angles.
(4y 2)o 66o 4y 64, y 16
(4y2)o
66o
(3x3)o and (2z)o are congruent alternate
interior angles.
(3x3)o 3(39) 3 114o
(2z)o 114o, z 57
25
Angles and Parallel Lines
Applications
Find the values of x, y and z in each figure.
There are other ways of doing this problem
correctly. If you worked it a different way,
would you be willing to share how you did it?
(2z)o
(3x3)o
(4y2)o
66o
26
Angles and Parallel Lines
Applications
Find the measures of all the angles on the object
if the measure of angle 1 is 30o.
Pause for time to work!
27
Angles and Parallel Lines
Applications
Find the measures of all the angles on the object
if the measure of angle 1 is 30o.
Perpendicular transversal theorem.
Perpendicular lines intersect in 4 right (90o)
angles.
90o
90o
90o
90o
90o
28
Angles and Parallel Lines
Applications
Find the measures of all the angles on the object
if the measure of angle 1 is 30o.
Vertical Angle
Vertical Angle
Alternate interior angle with angle 1.
30o
150o
150o
30o
90o
30o
Linear pairs are supplementary.
90o
30o
90o
Given
90o
90o
Vertical Angles
29
Angles and Parallel Lines
Applications
Find the measures of all the angles on the object
if the measure of angle 1 is 30o.
30o
Vertical angles.
150o
150o
30o
30o
90o
60o
60o
All angles have been found!
90o
30o
90o
90o
Since the transversal is , these two angles