1. Pearson Unit 1 Topic 3: Parallel & Perpendicular Lines 3-3: Proving Lines Parallel Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(5)(C) Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
(6)(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angle formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.

3. Basic Terms
Flow Proof—a form of proof in which arrows show the logical connections between the statements

4. What does this diagram tell you about lines n and m?
72 n m t

5. Postulate

6. 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.
Remember

7. What does this diagram tell you about lines n and m?
108 n m t

9. What does this diagram tell you about lines n and m?
72 n m
108 t

10. Theorem

11. What does this diagram tell you about lines n and m?
108 n m t

13. Proof #1 Given: ℓ || m Prove: 2  7 1. ℓ || m 1. Given 2. 2  6
Use the Converse of the Corresponding
Angles Postulate and the given to prove
the Alternate Exterior Angles Theorem.
Given: ℓ || m
Prove: 2  7
Statements
Reasons
1. ℓ || m
1. Given
2. 2  6
2. Corr. Angles Post.
3. 6  7
3. Vertical Angles Thm.
4. Transitive Prop. of 
4. 2  7

14. Proof #2 Use the Converse of the Corresponding
Angles Postulate and the given to show
that ℓ || m.
Given: m1 = m3
Prove: ℓ || m
Statements
Reasons
1. 1 & 3
corresponding angles
1. Def. of Corr. s
2. m1 = m 3
2. Given
3. 1  3
3. Def. of  s.
4. ℓ || m
4. Converse of Corr. s Post.

15. Proof #3
Refer to the diagram. Use the given information and the theorems you have learned to show that r || s.
Given: 4  8
Prove: r || s
Statements
Reasons
1. 4  8
1. Given
2. 4 and 8 are alt.
ext. s
2. Def. of alternate ext. s
3. Converse of Alt. Ext. s Thm.
3. r || s

16. Example: 4
Use the given information and the theorems you have learned to show that r || s.
Given: m2 = (10x + 8)°,
m6 = (25x –67)°, x = 5
Prove: r || s
m2 = 10x + 8 = 10(5) + 8 = 58
m6 = 25x – 67= 25(5) – 67 = 58
2 & 6 are alternate interior ’s
2  6
r || s by Converse of Alt. Int. s Theorem

17. Example: 5
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.

18. Example: 5 continued Thoughts:
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.

19. Example: 5 continued m1 = 8x + 20 = 8(15) + 20 = 140
m1+m2 = = 180
1 & 2 are same-side interior s
A || B by Converse of Same Side Int. s

20. Proof #6
Converse of the Alternate Exterior
Angles Theorem

21. EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW.
ALSO REMEMBER TO LOG-ON TO YOUR PEARSON ACCOUNT TO LOOK AT OTHER EXAMPLES BEFORE BEGINNING THE ON-LINE HW AND THE WRITTEN HW.

22. Example: 7
What if…? Suppose the corresponding angles on the same 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° y + 6 = 3(8) + 6 = 30°
The angles are congruent, so the oars are || by the
Converse of the Corr. s Post.