Proving Lines Parallel Lesson Presentation.ppt

Use the given information and the theorems you have learned to show that r || s.
A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.
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.
Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.
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
A line through the center of the horizontal piece forms a transversal to pieces A and B.
Use the given information and the theorems you have learned to show that r || s.
Use the given information and the theorems you have learned to show that r || s.
A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.
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.
Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.
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
A line through the center of the horizontal piece forms a transversal to pieces A and B.
Use the given information and the theorems you have learned to show that r || s.
Use the given information and the theorems you have learned to show that r || s.
A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.
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.
Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.
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
A line through the center of the horizontal piece forms a transversal to pieces A and B.
Use the given information and the theorems you have learned to show that r || s.
Use the given information and the theorems you have learned to show that r || s.
A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1= (8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.
Recall that the conver

1. Holt McDougal Geometry
3-3Proving Lines Parallel3-3Proving Lines Parallel
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry

2. Holt McDougal Geometry
3-3Proving Lines Parallel
Warm Up
State the converse of each statement.
1.If a= b, then a+ c= b+ c.
2.If mA+ mB= 90°, then Aand Bare
complementary.
3.If AB+ BC= AC, then A, B, and Care collinear.
If a+ c= b+ c, then a= b.
If Aand Bare complementary,
then mA+ mB=90°.
If A, B, andCare collinear, thenAB + BC= AC.

3. Holt McDougal Geometry
3-3Proving Lines Parallel
Use the angles formed by a transversal
to prove two lines are parallel.
Objective

4. Holt McDougal Geometry
3-3Proving Lines Parallel
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.

5. Holt McDougal Geometry
3-3Proving Lines Parallel

6. Holt McDougal Geometry
3-3Proving Lines Parallel
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ|| m.
Example 1A: Using the Converse of the
Corresponding Angles Postulate
4 8
4 and 8 are corresponding angles. Figure
ℓ|| m Conv. of Corr. s Post.
4 8 Given

7. Holt McDougal Geometry
3-3Proving Lines Parallel
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ|| m.
Example 1B: Using the Converse of the
Corresponding Angles Postulate
m3 = (4x–80)°,
m7 = (3x–50)°, x= 30
m3 = 4(30)–80 = 40 Subst./Simp. 30 for x.
m7 = 3(30) –50 = 40 Subst./Simp. 30 for x.
ℓ || m Conv. of Corr. s Post.
3 7 Def. of s.
m3 = m7 Trans. Prop. of Equality
3 & 7 are Corr. ’s Figure

8. Holt McDougal Geometry
3-3Proving Lines Parallel
Check It Out!Example 1a
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ|| m.
m1 = m3
1 3 Def. of Cong. Angles.
ℓ|| m Conv. of Corr. s Post.
1 and 3 are corresponding angles Figure

9. Holt McDougal Geometry
3-3Proving Lines Parallel
Check It Out!Example 1b
Use the Converse of the Corresponding Angles
Postulate and the given information to show
that ℓ|| m.
m7 = (4x+ 25)°,
m5 = (5x+ 12)°, x= 13
m7 = 4(13)+ 25 = 77 Subst./Simp. 13 for x.
m5 = 5(13) + 12 = 77 Subst./Simp. 13 for x.
ℓ || m Conv. of Corr. s Post.
7 5 Def. of s.
m7 = m5 Trans. Prop. of Equality
5 & 7 are Corr. ’s Figure

10. Holt McDougal Geometry
3-3Proving Lines Parallel

11. Holt McDougal Geometry
3-3Proving Lines Parallel

12. Holt McDougal Geometry
3-3Proving Lines Parallel
Use the given information and the theorems you
have learned to show that r|| s.
Example 2A: Determining Whether Lines are Parallel
4 8
4 8 Given
r|| s Conv. Of Alt. Ext. s Thm.
4 and 8 are alt.ext. ’sFigure

13. Holt McDougal Geometry
3-3Proving Lines Parallel
m2 = (10x+ 8)°,
m3 = (25x–3)°, x= 5
Use the given information and the theorems you
have learned to show that r|| s.
Example 2B: Determining Whether Lines are Parallel
m2 = 10x + 8
= 10(5) + 8 = 58 Subst./Simp. 5 for x.
m3 = 25x–3
= 25(5) –3 = 122 Subst./Simp. 5 for x.
2 and 3 are Same-side Int. ’sFigure

14. Holt McDougal Geometry
3-3Proving Lines Parallel
m2 = (10x+ 8)°,
m3 = (25x–3)°, x= 5
Use the given information and the theorems you
have learned to show that r|| s.
Example 2B Continued
r|| s Conv. of Same-Side Int. s Thm.
m2 + m3= 58°+ 122°= 180°Substitution

15. Holt McDougal Geometry
3-3Proving Lines Parallel
Check It Out!Example 2a
m1= m5
Refer to the diagram. Use the given information
and the theorems you have learned to show
that r || s.
1 and 5 are alternate exterior angles Figure
r|| s Conv. of Alt. Ext. s Thm.
15Def. Congruent ’s

16. Holt McDougal Geometry
3-3Proving Lines Parallel
Check It Out!Example 2b
Refer to the diagram. Use the given information
and the theorems you have learned to show
that r || s.
m3 = 2x, m7 = (x+ 50),
x= 50
m3 =m7 Trans. Prop =
3 7 Def. of Congr. angles
r||s Conv. of the Alt. Int. s Thm.
m3 = 2(50) = 100° Subst./Simp. 50 for x.
m7 = 50 + 50 = 100° Subst./Simp. 5 for x.
3 and 7 are alt. int. ’sFigure

17. Holt McDougal Geometry
3-3Proving Lines Parallel
Example 3: Proving Lines Parallel
Given:p || r , 1 3
Prove:ℓ || m

18. Holt McDougal Geometry
3-3Proving Lines Parallel
Example 3 Continued
Statements Reasons
1.p|| r
5.ℓ ||m
2.3 2
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.

19. Holt McDougal Geometry
3-3Proving Lines Parallel
Check It Out!Example 3
Given: 1 4, 3 and 4 are supplementary.
Prove:ℓ|| m

20. Holt McDougal Geometry
3-3Proving Lines Parallel
Check It Out!Example 3 Continued
Statements Reasons
1.1 4 1.Given
2.m1 = m4 2.Def. s
3.3 and4 are supp.3.Given
4.m3 + m4 = 1804.Def. of Supp. s
5.2 3 5.Vert. s Thm.
6.m2=m3 6. Def. s
7.m2 + m1 = 1807.Substitution
8.ℓ || m 8.Conv. of Same-Side
Interior sPost.

21. Holt McDougal Geometry
3-3Proving Lines Parallel
Example 4: Carpentry Application
A carpenter is creating a woodwork pattern
and wants two long pieces to be parallel.
m1= (8x + 20)°and m2 = (2x + 10)°. If
x = 15, show that pieces A and B are
parallel.

22. Holt McDougal Geometry
3-3Proving Lines Parallel
Example 4 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 xin each expression.

23. Holt McDougal Geometry
3-3Proving Lines Parallel
Example 4 Continued
m1 = 8x+ 20
= 8(15) + 20 = 140
m2 = 2x + 10
= 2(15) + 10 = 40
m1+m2 = 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.

24. Holt McDougal Geometry
3-3Proving Lines Parallel
Check It Out!Example 4
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. sPost.

25. Holt McDougal Geometry
3-3Proving Lines Parallel
Lesson Quiz: Part I
Name the postulate or theorem
that proves p || r.
1. 4 5 Conv. of Alt. Int. sThm.
2. 2 7 Conv. of Alt. Ext. sThm.
3. 3 7 Conv. of Corr. sPost.
4. 3 and 5 are supplementary.
Conv. of Same-Side Int. sThm.

26. Holt McDougal Geometry
3-3Proving Lines Parallel
Lesson Quiz: Part II
Use the theorems and given
information to prove p || r.
5. m2 = (5x + 20)°, m 7 = (7x + 8)°, and x = 6
m2 = 5(6) + 20 = 50°
m7 = 7(6) + 8 = 50°
m2 = m7, so 2 ≅7
p || r by the Conv. of Alt. Ext. sThm.

27. Holt McDougal Geometry
3-3Proving Lines Parallel