1. 3.3 Proving Lines are Parallel
Geometry
3.3 Proving Lines are Parallel

2. Geometry 3.3 Proving Lines are Parallel
Goals
Use postulates and theorems to prove two lines parallel.
Solve problems with parallel lines.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

3. Geometry 3.3 Proving Lines are Parallel
What we’ve been doing:
Given two parallel lines cut by a transversal…
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

4. Corresponding Angles Congruent
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

5. Alternate Exterior Angles Congruent
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

6. Alternate Interior Angles Congruent
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

7. Same Side Interior Angles Supplementary
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

8. Geometry 3.3 Proving Lines are Parallel
Now we want to
prove lines are
parallel.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

9. Geometry 3.3 Proving Lines are Parallel
Theorem:
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Converse:
If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

10. Geometry 3.3 Proving Lines are Parallel
In other words…
Corr. s   2 lines ||
(Theorem 3.5, Corr.  Converse)
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

11. Geometry 3.3 Proving Lines are Parallel
Theorem:
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Converse:
If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

12. Geometry 3.3 Proving Lines are Parallel
In other words…
Alt Int s   2 lines ||
(Theorem 3.6, Alternate Interior Angles Converse)
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

13. Geometry 3.3 Proving Lines are Parallel
Theorem:
If two parallel lines are cut by a transversal, then alternate Exterior angles are congruent.
Converse:
If two lines are cut by a transversal and alternate Exterior angles are congruent, then the lines are parallel.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

14. Geometry 3.3 Proving Lines are Parallel
In other words…
Alt Ext s   2 lines ||
(Theorem 3.7, Alternate Exterior Angles Converse)
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

15. Geometry 3.3 Proving Lines are Parallel
Theorem:
If two parallel lines are cut by a transversal, then same side interior angles are supplementary.
Converse:
If two lines are cut by a transversal and same side interior angles are supplementary, then the lines are parallel.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

16. Geometry 3.3 Proving Lines are Parallel
In other words… y°
x + y = 180° x°
SS Int s supp  2 lines ||
(Theorem 3.8, Same Side Interior Angles Converse)
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

17. To show two lines parallel, show that one of these is true:
Corresponding angles congruent.
Alternate interior angles congruent.
Alternate exterior angles congruent.
Same side interior angles supplementary.
You only need one pair for any one of these reasons.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

18. To show two lines parallel, show that one of these is true:
Corr. s 
Alt. Int. s 
Alt. Ext. s 
SS Int. s supp.
You only need one pair for any one of these reasons.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

19. Geometry 3.3 Proving Lines are Parallel
Example 1
Given: m  t, n  t.
Prove: m || n m n t 1
Not drawn to scale
Obviously 2
Proof:
Since m  t, 1 is a right angle.
Since n  t, 2 is a right angle.
All right angles are congruent, so 1  2.
This means m || n since alt int s   2 lines ||.
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

20. Geometry 3.3 Proving Lines are Parallel
Example 2
Given: 5  6; 6  4
Prove: A B C D 4 6 5
Proof:
If 5  6 and 6  4, then 5   Transitive Prop So
because of alt int s .
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

21. Geometry 3.3 Proving Lines are Parallel
Example 3
Find the value of x to make m || n. m n
These are alt int angles.
2x + 1 = 3x – 5
6 = x
(2x + 1)°
(3x – 5)°
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

22. Geometry 3.3 Proving Lines are Parallel
Write a proof
In the diagram, p || q and ∠1 is supplementary to ∠2. Prove r ||s using a proof.
Statements Reasons
∠1 is supp to ∠2
Given
m ∠1 + m ∠2 = 180
Def of supp ∠’s
p || q
Given
Alt Int ∠’s Theorem
m ∠ 2 = m ∠ 3
m ∠1 + m ∠ 3 = 180
Substitution
∠1 and ∠ 3 are supp
Def of supp ∠’s
r || s
Same Side Int ∠’s Converse
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

23. Theorem 3.9 Transitive Property of parallel lines
If two lines are parallel to the same line, then they are parallel to each other.
If m || n and p || n, then m || p. m n p
July 2, 2018
Geometry 3.3 Proving Lines are Parallel

24. Geometry 3.5 Using Properties of Parallel Lines
Slat 1 is parallel to slat 2.
Slat 2 is parallel to slate 3.
Why is slat 1 parallel to slat 3? 1 2 3
Theorem 3.9
If two lines are parallel to the same line, then they are parallel to each other.
July 2, 2018
Geometry 3.5 Using Properties of Parallel Lines