proving lines parallel · 2016-09-13 · holt mcdougal geometry proving lines parallel warm up...
TRANSCRIPT
![Page 1: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/1.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Warm Up State the converse of each statement.
1. If a = b, then a + c = b + c.
2. If mA + mB = 90°, then A and B are complementary.
3. If AB + BC = AC, then A, B, and C are collinear.
If a + c = b + c, then a = b.
If A and B are complementary, then mA + mB =90°.
If A, B, and C are collinear, then AB + BC = AC.
![Page 2: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/2.jpg)
Holt McDougal Geometry
Proving Lines Parallel
How can we prove lines are parallel?
Essential Question
Unit 1B Day 2
![Page 3: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/3.jpg)
Holt McDougal Geometry
Proving Lines Parallel
![Page 4: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/4.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
Example 1
Given: 4 8
4 8 4 and 8 are corresponding angles.
ℓ || m Conv. of Corr. s Post.
![Page 5: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/5.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 2
Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
Given: m1 = m3
1 3 1 and 3 are corresponding angles.
ℓ || m Conv. of Corr. s Post.
![Page 6: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/6.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
Example 3
Given: 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.
ℓ || m Conv. of Corr. s Post.
3 7 Def. of s.
m3 = m7 Trans. Prop. of Equality
![Page 7: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/7.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 4
Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
Given:
m7 = (4x + 25)°,
m5 = (5x + 12)°, x = 13
m7 = 4(13) + 25 = 77 Substitute 13 for x.
m5 = 5(13) + 12 = 77 Substitute 13 for x.
ℓ || m Conv. of Corr. s Post.
7 5 Def. of s.
m7 = m5 Trans. Prop. of Equality
![Page 8: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/8.jpg)
Holt McDougal Geometry
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.
![Page 9: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/9.jpg)
Holt McDougal Geometry
Proving Lines Parallel
![Page 10: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/10.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Use the given information and the theorems you have learned to show that r || s.
Example 5
Given: 4 8
4 8 4 and 8 are alternate exterior angles.
r || s Conv. Of Alt. Int. s Thm.
![Page 11: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/11.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 6
Given: m4 = m8
Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.
4 8 4 and 8 are alternate exterior angles.
r || s Conv. of Alt. Int. s Thm.
4 8 Congruent angles
![Page 12: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/12.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Given: m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5
Use the given information and the theorems you have learned to show that r || s.
Example 7
m2 = 10x + 8 = 10(5) + 8 = 58 Substitute 5 for x.
m3 = 25x – 3 = 25(5) – 3 = 122 Substitute 5 for x.
![Page 13: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/13.jpg)
Holt McDougal Geometry
Proving Lines Parallel
m2 = (10x + 8)°, m3 = (25x – 3)°, x = 5
Use the given information and the theorems you have learned to show that r || s.
Example 7 Continued
r || s Conv. of Same-Side Int. s Thm.
m2 + m3 = 58° + 122°
= 180° 2 and 3 are same-side interior angles.
![Page 14: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/14.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 8
Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.
Given:
m3 = 2x, m7 = (x + 50), x = 50
m3 = 100 and m7 = 100
3 7 r||s Conv. of the Alt. Int. s Thm.
m3 = 2x = 2(50) = 100° Substitute 50 for x.
m7 = x + 50 = 50 + 50 = 100° Substitute 5 for x.
![Page 15: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/15.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 9
Given: p || r , 1 3 Prove: ℓ || m
Statements Reasons
1. p || r
2. 3 2
5. ℓ ||m
3. 1 3
4. 1 2
2. Alt. Ext. s Thm.
1. Given
3. Given
4. Trans. Prop. of
5. Conv. of Corr. s Post.
![Page 16: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/16.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 10
Given: 1 4, 3 and 4 are supplementary.
Prove: ℓ || m
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.
![Page 17: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/17.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 11
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.
![Page 18: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/18.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 11 Continued
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.
![Page 19: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/19.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 11 Continued
m1 = 8x + 20
= 8(15) + 20 = 140
m2 = 2x + 10
= 2(15) + 10 = 40
m1+m2 = 140 + 40
= 180
Substitute 15 for x.
Substitute 15 for x.
1 and 2 are supplementary.
The same-side interior angles are supplementary, so pieces A and B are parallel by the Converse of the Same-Side Interior Angles Theorem.
![Page 20: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/20.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Example 12
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.
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.
![Page 21: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/21.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Assignment:
• pg 90-91 #2-10 even, 11, 12-22 even
![Page 22: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/22.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Lesson Quiz: Part I
Name the postulate or theorem that proves p || r.
1. 4 5 Conv. of Alt. Int. s Thm.
2. 2 7 Conv. of Alt. Ext. s Thm.
3. 3 7 Conv. of Corr. s Post.
4. 3 and 5 are supplementary.
Conv. of Same-Side Int. s Thm.
![Page 23: Proving Lines Parallel · 2016-09-13 · Holt McDougal Geometry Proving Lines Parallel Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. 2. If m A + m](https://reader035.vdocuments.net/reader035/viewer/2022080719/5f79831217fbb364a565800d/html5/thumbnails/23.jpg)
Holt McDougal Geometry
Proving Lines Parallel
Lesson Quiz: Part II
Use the theorems and given information to prove p || r.
5. m2 = (5x + 20)°, m 7 = (7x + 8)°, and x = 6
m2 = 5(6) + 20 = 50° m7 = 7(6) + 8 = 50° m2 = m7, so 2 ≅ 7
p || r by the Conv. of Alt. Ext. s Thm.