3-3 PROVING LINES PARALLEL CHAPTER 3. SAT PROBLEM OF THE DAY.
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Transcript of 3-3 PROVING LINES PARALLEL CHAPTER 3. SAT PROBLEM OF THE DAY.
![Page 1: 3-3 PROVING LINES PARALLEL CHAPTER 3. SAT PROBLEM OF THE DAY.](https://reader036.fdocuments.in/reader036/viewer/2022070403/56649f305503460f94c4b73e/html5/thumbnails/1.jpg)
3 - 3 P R OV I N G L I N E S PA RA L L E L
CHAPTER 3
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SAT PROBLEM OF THE DAY
• A cube is inscribe in a sphere of radius 6. What is the volume of the cube?• A)• B)• C)216• D)• E)
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OBJECTIVES
• Use the angles formed by a transversal to prove two lines are parallel.
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CONVERSE
• Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem.
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CONVERSE
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EXAMPLE#1
• Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
4 8
4 8 4 and 8 are corresponding angles.
ℓ || m Conv. of Corr. s Post.
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EXAMPLE#2
• Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
• m3 = m7 Trans. Prop. of Equality• 3 7 Def. of s.• ℓ || m Conv. of Corr. s
m3 = (4x – 80)°, m7 = (3x – 50)°, x = 30
m3 = 4(30) – 80 = 40 Substitute 30 for x.
m7 = 3(30) – 50 = 40 Substitute 30 for x.
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STUDENT GUIDED PRACTICE
• Do problems 1-3 in the book page 166
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PROVING LINES PARALLEL
The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.
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THEOREMS
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EXAMPLE
• Use the given information and the theorems you have learned to show that r || s.
4 8
4 8 4 and 8 are alternate exterior angles.
r || s Conv. Of Alt. Ext. s Thm.
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EXAMPLE
• Use the given information and the theorems you have learned to show that r || s.
• m3 = 25x – 3• = 25(5) – 3 = 122Substitute 5 for x.
m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5
m2 = 10x + 8 = 10(5) + 8 = 58 Substitute 5 for x.
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CONTINUE EXAMPLE
• m2 + m3 = 58° + 122°• = 180° 2 and 3 are same-side
interior angles.• r || s Conv. of Same-Side Int. s Thm.
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STUDENT GUIDED PRACTICE
• Do problems 4-6 in your book page 166
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PROVING PARALLEL LINES
• Given: p || r , 1 3• Prove: ℓ || m
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SOLUTION
• statements reasons
4. 1 2
1. p || r
5. ℓ ||m
2. 3 2
3. 1 3
2. Alt. Ext. s Thm.
1. Given
3. Given
4. Trans. Prop. of
5. Conv. of Corr. s Post.
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EXAMPLE
• Given: 1 4, 3 and 4 are supplementary.• Prove: ℓ || m
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SOLUTION
Statements Reasons
1. 1 4 1. Given
2. m1 = m4 2. Def. s
3. 3 and 4 are supp. 3. Given
4. m3 + m4 = 180 4. Trans. Prop. of 5. m3 + m1 = 180 5. Substitution
6. m2 = m3 6. Vert.s Thm.
7. m2 + m1 = 180 7. Substitution
8. ℓ || m 8. Conv. of Same-Side Interior s Post.
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APPLICATION
• A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.
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• A line through the center of the horizontal piece forms a transversal to pieces A and B.• 1 and 2 are same-side interior angles. If 1 and
2 are supplementary, then pieces A and B are parallel.• Substitute 15 for x in each expression.
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• m1 = 8x + 20• = 8(15) + 20 = 140• m2 = 2x + 10• = 2(15) + 10 = 40• m1+m2 = 140 + 40• = 180• The same-side interior angles are supplementary,
so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem.
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APPLICATION
• What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where • y = 8. Show that the oars are parallel.
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CONTINUE
• 4y – 2 = 4(8) – 2 = 30° 3y + 6 = 3(8) + 6 = 30°• The angles are congruent, so the oars are || by
the Conv. of the Corr. s Post.
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HOMEWORK!!!
• Do problems 12-18 and problem 22 in your book page 166 and 167
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CLOSURE
• Today we learned about parallel lines • Next class we are going to learned about
perpendicular lines