Warm‐Up
1. True or false: Each of the three lines is a transversal for the other two.
2. Name 4 pairs of corresponding angles.
3. Name 4 pairs of alternate exterior angles.
4. Name 4 pairs of alternate interior angles.
5. Name 4 pairs of consecutive interior angles.
1
Homework 3.1
4. BC
6. plane ABFE
8. PQ,PN
10. No, there is no right angle symbol to indicate they are perpendicular.
12. ∠3 and ∠6,∠4 and ∠5
14. ∠3 and ∠5,∠4 and ∠6
18. Corresponding
20. Consecutive interior
22. Alternate interior
24. Always, If two lines are parallel then they are ALWAYS coplanar.
26. Sometimes, If three lines intersect at one point then they are SOMETIMES coplanar.
46. A serving of soup is low in sodium
48. m∠1 = 70, m∠2 = 110, m∠3 = 70
2
3
3.2 Use Parallel Lines and
Transversals
4
2
1
3
4
6
5
7
8
5
1
2
3
4
6
5
7
8
6
1
2
3
4
5
6
7
8
7
2
1
3
4
6
5
7
8
8
1
2
3
4
6
5
7
8
9
What are parallel lines?
What are transversals?
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Where do Parallel Lines Cut by
Transversals Exist?
• Parking spots
• Bookshelves
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12
Parallel lines and Transversals
We know about the special angle pairs that are
formed by transversals.
Now will see what happens when the transversal
crosses two parallel lines
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Protractors!
80
90
100
110
70
60
110
50
100
90
150
130
140
50
130
140
30
70
60
120
40
120
80
0°
40
150
30
160
20
160
10
0
20
170
10
180
0
170
180
14
110
120
130
140
100
90
80
80
40
160
170
10
31°
120
150
50
20
100
110
60
60
30
90
70
70
180
0
50
130
40
140
150
30
20
160
170
10
180
0
15
0
10
20
180
30
170
160
40
150
50
140
130
60
120
70
110
80
100
90°
90
90
80
100
70
110
60
120
50
40
130
30
0
10
140
20
150
160
180
170
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Activity!
1. Draw a transversal to your two parallel lines don't make it perpendicular (USE THE RULER!!!)
2. Number your angles.
3. Measure each angle with the protractor.
4. Make conjectures about (WRITE THEM DOWN):
Alternate interior Angles
Alternate exterior Angles
Corresponding Angles
Consecutive Interior Angles
Consecutive Exterior Angle
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What conjectures did you make about the angles?
a. Alternate interior:
b. Alternate Exterior:
c. Corresponding:
d. Consecutive interior:
e. Consecutive exterior
19
• Notice EVERYONE’s diagram is different, yet
we all got the same angles to be congruent.
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Postulate 15: Corresponding Angles
Postulate
If two parallel lines are cut by a transversal,
then the pairs of corresponding angles are
congruent.
1
5
∠1≅∠5
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Theorem 3.1: Alternate Interior
Angles Theorem
If two parallel lines are cut by a transversal, then
the pairs of alternate interior angles are
congruent.
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Prove!!! Alternate Interior Angles
are Congruent
Given: a||b and c is a transversal.
Prove: ∠ 3≅∠6 and ∠4 ≅∠5
c
1 2
3 4
5 6
7 8
a
b
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Given: a||b and c is a transversal.
Prove: ∠ 3≅∠6 and ∠4 ≅∠5
1 2
3 4
5 6
7 8
c
Statement
Reasons
1. a||b and c is a transversal. 2. ∠ 4 ≅∠ 8 3. ∠ 7 ≅∠ 3
4. ∠ 8 ≅∠ 5 ∠ 6 ≅∠ 7 5. ∠ 4 ≅∠ 5 6. ∠ 3 ≅∠ 6
1. Given
2. corresponding angles post.
3. corresponding angles post.
4. vertical angles ≅ a
b
5. Transitive 2,4
6. Transitive 3,5 24
Theorem 3.2: Alternate Exterior
Angles Theorem
If two parallel lines are cut by a transversal, then
the pairs of alternate exterior angles are
congruent.
2
∠2≅∠7
7
25
What types of angles are these?
How do we solve for x?
26
Theorem 3.3: Consecutive Interior
Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles are
supplementary.
4
6
m∠4+m∠6=180°
27
What types of angles are these?
How do we solve for x?
28
Use the diagram
1. If m∠1=105°, find m∠4, m∠5, and m∠8. Tell
which postulate or theorem you use in each
case.
1 2
3 4
5 6
7 8
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Use the diagram
2. If m∠3=68° and m m∠8=(2x+4) °, what is the
value of x? Show your steps!
1 2
3 4
5 6
7 8
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Determining Angle Measures
Given: m∠ 5 = 130° find m∠4, m∠5,
m∠7, and m∠6.
c
1 2
3 4
5 6
7 8
a
b
31
l
m
a
7
8
9
2
b
1
3
4
5
6
• Given l||m, m 1 = 98, and m 2 = 40
• Find the measure of the following angles (explain)
• 3
• 4
• 5
• 6
• 7
• 8
• 9
32
l
m
a
7
8
9
2
b
1
3
4
5
6
• Given l||m, m 1 = 98, and m 2 = 40
• Find the measure of the following angles (explain)
•
•
•
•
•
•
•
3= (180‐98)=82 linear pair
4 =98 corresponding angles
5= (180‐98) = 82 linear pair
6 = (180‐40) = 140 linear pair
7 = 40 vertical angles
8 = 40 alternate interior angles
9 = 140 alternate exterior angles
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Find the value of x and y
34
Given: l
||m, t||r
Prove: ∠ 1≅∠2
Statements
Reasons
35
Parallel Lines
• No matter what our parallel lines or
transversal looks like the properties will hold.
Properties of Parallel lines
36
Homework
• Pg. 157: 4‐19, 22‐24, 30‐37, 41
• #37 is proving theorem 3.2!
• #41 is proving theorem 3.3!
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