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Angles and Parallel
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Geometry D – Section 3.2
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.
Angles and Parallel Lines
Extend your lines the full height and width of the paper.
Pause for time to work!
Angles and Parallel Lines
Label the angles as shown below.
1
2
3 4
5 6
7 8
Pause for time to work!
Angles and Parallel Lines
Measure all angles using a
protractor to the nearest
degree.
1
3
2
4
5 6
7 8
Pause for time to work!
Angles and Parallel Lines
Measure all angles using a
protractor to the nearest
degree.
127o
53o
1
53o 3
127o
5 6
53o7
8
127o
53o
2
4
127o
Note: Your measurements
may be different values but
should be in the same
pattern.
Angles and Parallel Lines
127o
53o
1
53o 3
127o
5 6
53o7
8
127o
53o
2
4
127o
Identify the relationship
between the following angles?
1,  2
3, 4
5, 6
7, 8
From Chapter 2, the
angles are linear pairs.
What can be said about the measures of the linear pairs?
Linear pairs are supplementary
(sum to 180o).
Angles and Parallel Lines
127o
53o
1
53o 3
127o
5 6
53o7
8
127o
53o
2
4
127o
Identify the relationship
between the following angles?
1, 4
2, 3
5, 8
6, 7
From Chapter 2, the
angles are vertical angles.
What can be said about the measures of the vertical
angles?
Vertical angles are congruent angles.
Angles and Parallel Lines
127o
53o
1
53o 3
127o
5 6
53o7
8
127o
53o
2
4
127o
Identify the relationship
between the following angles?
1, 5
2, 6
3, 7
4, 8
The angles are
corresponding angles.
What can be said about the measures of the corresponding
angles?
The measures are equal and the angles are
congruent.
Angles and Parallel Lines
Corresponding Angles Postulate –
If two parallel lines are cut by a transversal, then
each pair of corresponding angles are congruent.
Angles and Parallel Lines
Identify the relationship
between the following angles?
127o
53o
1
53o 3
2
4
127o
3, 6
4, 5
127o
5 6
53o7
8
127o
53o
The angles are alternate
interior angles.
What can be said about the measures of the alternate
interior angles?
The measures are equal and the angles are
congruent.
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!
Angles and Parallel Lines
Identify the relationship
between the following angles?
127o
53o
1
53o 3
2
4
127o
3, 5
4, 6
127o
5 6
53o7
8
127o
53o
The angles are alternate
interior angles.
What can be said about the measures of the alternate
interior angles?
The measures add to 180o.
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!
Angles and Parallel Lines
Identify the relationship
between the following angles?
127o
53o
1
53o 3
2
4
127o
1, 8
2, 7
127o
5 6
53o7
8
127o
53o
The angles are alternate
exterior angles.
What can be said about the measures of the alternate
interior angles?
The measures are equal and the angles are
congruent.
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:
1  8, 2  7
p || q, t is a
transversal of p & q
t
3
5
7 8
1
2
p
4
6
Statement
q
Reason
?
Given
1  5, 2  6
Corresponding  ‘s
?
are 
5  8, 6  7
?
Vertical
1  8, 2  7
‘s are 
Transitive
? Property
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!
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.
Angles and Parallel Lines
Applications –
k
1
Given j || k,
m1  43o , m14  24o
Makethe
a sketch
of of
the
Find
measure
problem in your notes.
1. 3  43o
Corresponds with 1.
j
3
2
6 7
4
8
5
9
11
10
2.
12
13
9  24o
Alternate exterior with
14.
3.
14
10  156o
Linear pair with 9.
180o – 24o = 156o
Angles and Parallel Lines
Applications –
Given j || k,
m1  43o , m14  24o
Find the measure of
k
1
j
3
2
6 7
4
8
5
9
4.
4  137o
Linear pair with 3.
11
10
5. 11  156o
Vertical angle with
10.
12
13
6.
14
7 
43o
Vertical  with  1.
Alternate Interior of 3.
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:
mDCA  (5 x  2)o
D
mACB  (9 x  10)
C
B
A
o
mDBE  (3 y  1)o
E
Pause for time to work!
Angles and Parallel Lines
Applications –
Given:
o
mDCA  (5 x  2)
mACB  (9 x  10)o
mDBE  (3 y  1)o
D
Linear
Pair
62o
C 118o
B
Solution
mDCA  mACB  180o
Linear pairs are supplementary.
(5x + 2) + (9x + 10) = 180o
14x + 12 = 180
14x = 168
x = 12
mDCA  5(12)  2  62o
mACB  9(12)  10  118o
By corresponding angles,
mDBE  mDCA
A
62o
E
3y – 1 = 62
3y = 63
o
m

DBE

62
y = 21 and
Angles and Parallel Lines
Applications –
Find the values of x, y and z in each figure.
B
(2z)o
A
(4y+2)o
(3x–3)o
66o C
D
Pause for time to work!
Angles and Parallel Lines
Applications –
B
o
(2z)A
(4y+2)o
D
o
66o (3x–3)
66o C
ABC is a corresponding
angle with the angle of 66o.
(3x – 3)o and 66o are linear
pairs and sum to 180o.
(3x – 3)o + 66o = 180o
3x + 63 = 180
3x = 117, x = 39
(4y + 2)o and 66o are congruent
alternate interior angles.
(4y + 2)o = 66o
4y = 64, y = 16
(3x–3)o and (2z)o are congruent
alternate interior angles.
(3x–3)o = 3(39) – 3 = 114o
(2z)o = 114o, z = 57
Angles and Parallel Lines
Applications –
Find the values of x, y and z in each figure.
B
(2z)oA
(4y+2)o
D
(3x–3)o
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?
66o C
Angles and Parallel Lines
Applications –
Find the measures of all the angles on the
object if the measure of angle 1 is 30o.
1
Pause for time to work!
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.
1
90o
90o
90o
o
90
90o
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
30o
150o
150o
1
Linear pairs
are supplementary.
Given
Vertical
Angle
30o
Alternate interior
angle with angle 1.
30o
90o
90o
90o
o
90
90o
Vertical
Angles
30o
Angles and Parallel Lines
Applications –
Find the measures of all the angles on the
object if the measure of angle 1 is 30o.
30o
150o
150o
1
30o
Vertical angles.
30o
90o
60o
All angles have
been found!
90o
90o
o
90
90o
60o
30o
Since the transversal is  ,
these two angles must add
to 90o using angle addition.
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!!!!