1. Geometry Angles formed by Parallel Lines and Transversals
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2. Give an example of each angle pair.
Warm Up
Give an example of each angle pair.
1) Alternate interior angles
2) Alternate exterior angles
3)Same side interior angles
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3. Parallel, perpendicular and skew lines
When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed.
1 2
3 4
5 6
7 8
There are several special pairs of angles formed from this figure.
Vertical pairs:
Angles 1 and 4  Angles 2 and 3  Angles 5 and 8  Angles 6 and 7
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4. Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3
Supplementary pairs:
Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3
Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7
1 2
3 4
5 6
7 8
Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are three other special pairs of angles. These pairs are congruent pairs.
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5. Corresponding angle postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. p q t
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6. Using the Corresponding angle postulate
Find each angle measure.
A) m( ABC) x0 B C A
x = 80
corresponding angles
m( ABC) = 800
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7. subtract x from both sides
B) m( DEF)
(2x-45)0 = (x+30)0
corresponding angles
x – 45 = 30
subtract x from both sides
x = 75
add 45 to both sides
m( DEF) = (x+30)0
= (75+30)0
= 1050
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8. Now you try!
1) m( DEF) R x0 S
1180 Q
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9. Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles postulate is given as a postulate, it can be used to prove the next three theorems.
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10. Alternate interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent.
Hypothesis
Conclusion
1 2
4 3
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11. Alternate exterior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate exterior angles are congruent.
Hypothesis
Conclusion
5 6
8 7
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12. Same-side interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Same-side interior angles are supplementary.
Hypothesis
Conclusion
1 2
4 3
m m 4 =1800
m m 3 =1800
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13. Alternate interior angles theorem1 2 3 m l
Given: l || m
Prove:
Proof:
l || m
Given
Corresponding angles
Vertically opposite angles
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14. Finding Angle measures
Find each angle measure.
A) m( EDF)
1250 B C A x0 D E F
x = 1250
m( DEF) = 1250
Alternate exterior angles theorem
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15. R T B) m( TUS) S U 36x = 180 x = 5 m( TUS) = 23(5)0 = 1150 13x0 23x0
Same-side interior angles theorem
36x = 180
Combine like terms
x = 5
divide both sides by 36
m( TUS) = 23(5)0
Substitute 5 for x
= 1150
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16. 2) Find each angle measure.
Now you try!
2) Find each angle measure. B C E D
(2x+10)0 A
(3x-5)0
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17. By the Alternative Exterior Angles Theorem, (25x+5y)0 = 1250
A treble string of grand piano are parallel. Viewed from above, the bass strings form transversals to the treble string. Find x and y in the diagram.
(25x+5y)0
(25x+4y)0
1200
1250
By the Alternative Exterior Angles Theorem, (25x+5y)0 = 1250
By the Corresponding Angles Postulates, (25x+4y)0 = 1200
(25x+5y)0 = 1250
- (25x+4y)0 = 1200
y = 5
25x+5(5) = 125
x = 4, y = 5
Subtract the second equation from the first equation
Substitute 5 for y in 25x +5y = 125. Simplify and solve for x.
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18. 3) Find the measure of the acute angles in the diagram.
Now you try!
3) Find the measure of the acute angles in the diagram.
(25x+5y)0
(25x+4y)0
1200
1250
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19. Find each angle measure:
Assessment
Find each angle measure:
1) m( JKL)
2) m( BEF)
(7x-14)0
(4x+19)0 G A B C F D H E
1270 x0 K J L
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20. Find each angle measure:1
(3x+9)0
6x0 A B C D Y X E Z
4) m( CBY)
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21. Find each angle measure:
5) m( KLM)
1150 Y0 K M L
6) m( VYX) Y X W Z
(2a+50)0 V
4a0
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22. State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures: 1 2 3 4 5
7) m 1 = (7x+15)0 , m 2 = (10x-9)0
8) m 3 = (23x+15)0 , m 4 = (14x+21)0
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23. Parallel, perpendicular and skew lines
Let’s review
Parallel, perpendicular and skew lines
When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed.
1 2
3 4
5 6
7 8
There are several special pairs of angles formed from this figure.
Angles 1 and 4  Angles 2 and 3  Angles 5 and 8  Angles 6 and 7
Vertical pairs:
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24. Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3
Supplementary pairs:
Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3
Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7
1 2
3 4
5 6
7 8
Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are three other special pairs of angles. These pairs are congruent pairs.
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25. Corresponding angle postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. p q t
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26. Using the Corresponding angle postulate
Find each angle measure.
A) m( ABC) x0 B C A
x = 80
corresponding angles
m( ABC) = 800
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27. subtract x from both sides
B) m( DEF)
(2x-45)0 = (x+30)0
corresponding angles
x – 45 = 30
subtract x from both sides
x = 75
add 45 to both sides
m( DEF) = (x+30)0
= (75+30)0
= 1050
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28. Alternate interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent.
Hypothesis
Conclusion
1 2
4 3
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29. Alternate exterior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate exterior angles are congruent.
Hypothesis
Conclusion
5 6
8 7
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30. Same-side interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Same-side interior angles are supplementary.
Hypothesis
Conclusion
1 2
4 3
m m 4 =1800
m m 3 =1800
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31. Alternate interior angles theorem1 2 3 m l
Given: l || m
Prove:
Proof:
l || m
Given
Corresponding angles
Vertically opposite angles
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32. You did a great job today!
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