1. 3.4 Proving Lines are Parallel

2. If two boats are sailing at a 45 degree angle to the wind and the wind is constant, will the two boats paths ever cross?
This is going to be how we will start off class.
This is a real life application problem for the students. I’m going to leave the part under the question empty because I want the class to have to develop a picture of what they think this will look like. I will take suggestions from the class and draw what they say, or have a student come up and do the same but be the scribe for the class. This will hopefully lead into some discussions and different answers.
The end result will be no, because corresponding angles are congruent, the boats’ paths are parallel and parallel lines don’t intersect.
This is because of the Corresponding Angles Converse which will be explained on the next slide.

3. Converses… Corresponding Angle Postulate Corresponding Angles Converse
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Corresponding Angles Converse
If two lines are cut by a transversal so that the corresponding angles are congruent then the lines are parallel
The first part of this section is talking about the converses of the postulate and theorems addressed yesterday. The students know what a converse is so I’m going to refresh their memory with the postulate or theorem and see if we can come up with the converse on our own.
So the Converse will slide in at a different time so we can talk about it, and then see how close the class was.
I will use the picture to give the students a way to visualize the difference.
The following is just an example that the students could address:
In the original you’re given the two lines are parallel and then you’re told that means <b and <f are congruent.
In the converse you’re given that <b and <f are congruent which means the two lines are parallel.

4. Decide which, if any, of the rays are parallel.62 58 59 61 A B C D E F G
This is using the corresponding angle converse as well.
This problem will be put up on the board and the students will work it out by themselves and then work with a partner and then we will come together as a class.
The students will find that line EB and line FD are not parallel but AE and CF are because they have the same corresponding angles, <AEF and <CFG.

5. Converses… Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Alternate Interior Angles Converse
If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.
The students know what a converse is so I’m going to refresh their memory with the postulate or theorem and see if we can come up with the converse on our own.
So the Converse will slide in at a different time so we can talk about it, and then see how close the class was.
I will use the picture to give the students a way to visualize the difference.
The following is just an example that the students could talk about:
In the original you’re given the two lines are parallel and then you’re told that means <c and <f are congruent.
In the converse you’re given that <c and <f are congruent which means the two lines are parallel.

6. Proof of the Alternate Interior Angle Converse3 m
Statements
Reasons 2 n 1
Given: <1 is congruent to <2
Prove: m is parallel to n
Here is just a simple proof of the Alternate Interior Angle Converse. The others are left to prove for homework.
The students will be given a few minutes to do this on their own, and then they will be asked to compare answers with a neighbor.
Then we will come back and complete it was a class.

7. Find the value of x that makes line j parallel to line k.
This is a problem is the introduction to the Consecutive interior angle converse and in incorporates some algebra.
The students will do this problem on their own, check their answer with a neighbor, then we will come together as a class and work it out on the board. I want them to think of why this is true. Once we discuss the answers we will then introduce the official reason, the Consecutive interior angle converse and it’s definition.
Answer: x has to equal 36 degrees k

8. Converses… Consecutive Interior Angles
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
Consecutive Interior Angles Converse
If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel
The students know what a converse is so I’m going to refresh their memory with the postulate or theorem and see if we can come up with the converse on our own.
So the Converse will slide in at a different time so we can talk about it, and then see how close the class was.
I will use the picture to give the students a way to visualize the difference.
The following is just an example that the students could talk about:
In the original you’re given the two lines are parallel and then you’re told that means <d and <f are supplementary.
In the converse you’re given that <d and <f are supplementary which means the two lines are parallel.

9. Converses… Alternate Exterior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate exterior angles are congruent then the lines are parallel
The students know what a converse is so I’m going to refresh their memory with the postulate or theorem and see if we can come up with the converse on our own.
So the Converse will slide in at a different time so we can talk about it, and then see how close the class was.
I will use the picture to give the students a way to visualize the difference.
The following is just an example that the students could talk about:
In the original you’re given the two lines are parallel and then you’re told that means <a and <h are supplementary.
In the converse you’re given that <a and <h are supplementary which means the two lines are parallel.

10. 3.5 Using Properties of Parallel Lines

11. Given: Line m is parallel to line n, and line n is parallel to line k
Prove: line m is parallel to line k
Statements
Reasons 1
This proof will be worked out as a class after the students try to figure it out on their own. This proof will lead to the theorem that is to follow. m 2 n 3 k

12. Theorem 3.11 Parallel Lines
If two lines are parallel to the same line then they are parallel to each other.
Theorem 3.12
Perpendicular Lines
If two lines are perpendicular to the same line, then they are parallel to each other.
The first theorem on this slide is what the students would have just discovered with the previous activity and the second theorem is conceptually the same thing.

13. Constructing Parallel Lines
Draw points Q and R on m. Draw PQ.
Draw an arc with the compass point at Q so that it crosses QP and QR.
Copy <PQR on QP. Be sure the two angles are corresponding. Label the new angle <TPS.
Draw PS.
Why are PS and QR parallel?
This construction will be done in class, and I will do it with the students using the projector that projects what I’m doing at the desk on to the promethean board.
Because <TPS and <PQR are congruent corresponding angles PS and QR are parallel.
This is a fun activity to break up the monotony of just copying notes and working problems.

14. What did you learn this lesson?
Can you prove lines p and q are parallel? Why or why not?
a b.
c d.
e. f. p q p q p q
105 p q 62
This is the review of the lesson that will be turned into me at the end of the period. This is how I will see if the students are grasping the material, and I can see if they have any questions 62 75
If you have any questions about this lesson or a previous one please put them at the end. p q r