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

2. Holt McDougal Geometry
3-3 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B = 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 mA + mB =90°.
If A, B, and C are collinear, then AB + BC = AC.

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

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

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

11. Holt McDougal Geometry
3-3 Proving Lines Parallel

12. Holt McDougal Geometry
3-3 Proving 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. ’s Figure

13. Holt McDougal Geometry
3-3 Proving 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. ’s Figure

14. Holt McDougal Geometry
3-3 Proving 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
3-3 Proving 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  5 Def.  Congruent ’s

16. Holt McDougal Geometry
3-3 Proving 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. ’s Figure

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

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

20. Holt McDougal Geometry
3-3 Proving 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 s Post.

21. Holt McDougal Geometry
3-3 Proving 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
3-3 Proving 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 x in each expression.

23. Holt McDougal Geometry
3-3 Proving 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
3-3 Proving 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. s Post.

25. Holt McDougal Geometry
3-3 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.

26. Holt McDougal Geometry
3-3 Proving 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. s Thm.

27. Holt McDougal Geometry
3-3 Proving Lines Parallel