Angles and Parallel Lines

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Angles and Parallel Lines

• Geometry D Section 3.2

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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.
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Angles and Parallel Lines
Extend your lines the full height and width of
the paper.
Pause for time to work!
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Angles and Parallel Lines
Label the angles as shown below.
1
2
3
4
6
5
7
8
Pause for time to work!
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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!
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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
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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).
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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.
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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.
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Angles and Parallel Lines
Corresponding Angles Postulate
If two parallel lines are cut by a transversal,
then each pair of corresponding angles are
congruent.
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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.
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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!
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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 add to 180o.
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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!
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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.
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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
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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!
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Angles and Parallel Lines
Applications
Gather into groups of not more than 3. Work the
following problems in your group.Compare your
answers to those provided.
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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
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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
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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!
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Angles and Parallel Lines
Applications Solution
Given
Linear pairs are supplementary.
(5x 2) (9x 10) 180o
14x 12 180
14x 168
x 12
LinearPair
By corresponding angles,
62o
118o
3y 1 62
62o
3y 63
y 21 and
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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!
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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 180o3x 63 1803x 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
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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
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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!
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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
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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
VerticalAngle
Alternate interior angle with angle 1.
30o
150o
150o
30o
90o
30o
Linear pairsare supplementary.
90o
30o
90o
Given
90o
90o
VerticalAngles
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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
must add to 90o using angle addition.
90o
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Angles and Parallel Lines
Assignment
3.2 / 17-20, 24, 26, 30, 32, 36, 38, 40,
43, 48, 55, 57, 60, 62
Please return your protractor!!!! Thank you Mr.
Matzke!!!!