Angles and Parallel Lines
Obtain a ½ sheet of graph paper and a protractor. Construct two || lines and a transversal similar to the image on the next slide.
We are going to investigate the relationship of various angles created by two parallel lines and a transversal.
Angles and Parallel Lines
Extend your lines the full height and width of the paper.
Pause for time to work!
Angles and Parallel Lines
Measure all angles using a protractor to the nearest degree.
1 2
3 4
657 8
Pause for time to work!
Angles and Parallel Lines
Measure all angles using a protractor to the nearest degree.
1 2
3 4
657 8
127o
Note: Your measurements may be different values but should be in the same pattern.
127o
127o
127o
53o
53o
53o
53o
Angles and Parallel LinesIdentify the relationship between the following angles?
1 2
3 4
657 8
127o
From Chapter 2, the angles are linear pairs.
127o
127o
127o
53o
53o
53o
53o
1, 2
3, 4
5, 6
7, 8
What can be said about the measures of the linear pairs?
Linear pairs are supplementary (sum to 180o).
Angles and Parallel LinesIdentify the relationship between the following angles?
1 2
3 4
657 8
127o
From Chapter 2, the angles are vertical angles.
127o
127o
127o
53o
53o
53o
53o
1, 4
2, 3
5, 8
6, 7
What can be said about the measures of the vertical angles? Vertical angles are congruent angles.
Angles and Parallel LinesIdentify the relationship between the following angles?
1 2
3 4
657 8
127o
The angles are corresponding angles.
127o
127o
127o
53o
53o
53o
53o
1, 5
2, 6
3, 7
4, 8
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?
1 2
3 4
657 8
127o
The angles are alternate interior angles.
127o
127o
127o
53o
53o
53o
53o
3, 6
4, 5
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?
1 2
3 4
657 8
127o
The angles are alternate interior angles.
127o
127o
127o
53o
53o
53o
53o
3, 5
4, 6
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?
1 2
3 4
657 8
127o
The angles are alternate exterior angles.
127o
127o
127o
53o
53o
53o
53o
1, 8
2, 7
What can be said about the measures of the alternate interior angles? The measures are equal and the angles are
congruent.
1 8, 2 7 Statement Reason
Angles and Parallel Lines
1 5, 2 6
5 8, 6 7
1 8, 2 7
Alternate Exterior Angles Theorem –
If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.
t
p
q
1 23 4
65
7 8
p || q, t is a transversal of p & q
Given
Prove:
Corresponding ‘s are
Vertical ‘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.
You will prove this theorem as a homework problem!
p
q
t
If t is perpendicular ( ) to p, then it is also perpendicular to q.
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.
Make a sketch of the problem in your notes.
Given j || k, Angles and Parallel Lines
jk
Applications –
1 2 3 4 5
10
6 7 8 911
1213
14
1 43 , 14 24o om m
3 1.
Corresponds with 1.
43o
9 2.Alternate exterior with 14.
24o
10 3.Linear pair with 9.180o – 24o = 156o
156o
Find the measure of
jk
Given j || k,
Find the measure of
Angles and Parallel Lines
Applications –
1 2 3 4 5
10
6 7 8 911
1213
14
1 43 , 14 24o om m
4 4.
Linear pair with 3.
137o
11 5.Vertical angle with 10.
156o
7 6.Vertical with 1.Alternate Interior of 3.
43o
D
C A
EB
Find the values of x and y in each figure.Find the measure of each given angle.
Note: Figures are not drawn to scale.
Angles and Parallel Lines
(9 10)om ACB x
Applications –
(5 2)om DCA x Given:
(3 1)om DBE y
Pause for time to work!
D
C A
EB
Angles and Parallel Lines
Applications – Solution
(5 2)om DCA x Given:
(9 10)om ACB x (3 1)om DBE y
180om DCA m ACB Linear pairs are supplementary.
(5x + 2) + (9x + 10) = 180o
14x + 12 = 18014x = 168x = 12
5(12) 2 62om DCA 9(12) 10 118om ACB
By corresponding angles,m DBE m DCA 3y – 1 = 623y = 63 y = 21 and 62om DBE
LinearPair
62o
118o
62o
D
A
B
C
Find the values of x, y and z in each figure.
Angles and Parallel Lines
Applications –
66o
(3x–3)o(2z)o
(4y+2)o
Pause for time to work!
D
A
B
C
is a corresponding angle with the angle of 66o.Angles and Parallel Lines
Applications –
ABC
66o
(3x–3)o(2z)o
(4y+2)o
66o
(3x – 3)o and 66o are linear pairs and sum to 180o.(3x – 3)o + 66o = 180o
3x + 63 = 1803x = 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
D
A
B
C
Find the values of x, y and z in each figure.
Angles and Parallel Lines
Applications –
66o
(3x–3)o(2z)o
(4y+2)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?
1
Angles and Parallel Lines
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
Applications –
Pause for time to work!
1
Angles and Parallel Lines
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
Applications –
90o
90o90o
Perpendicular lines intersect in 4 right (90o) angles. 90o
90o
Perpendicular transversal theorem.
1Linear pairsare supplementary.
Angles and Parallel Lines
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
Applications –
30o
90o90oGiven
90o
90o
30o
Vertical Angle
90o
150o
150o
VerticalAngle
Alternate interior angle with angle 1.
30o
30o
VerticalAngles
1
Angles and Parallel Lines
Find the measures of all the angles on the object if the measure of angle 1 is 30o.
Applications –
30o
90o90o
90o
90o
30o
90o
150o
150o
30o
30o
Since the transversal is , these two angles must add to 90o using angle addition.
60o60o
Vertical angles.
All angles have been found!
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