More Proofs!

Warm-up

1. What do you need to know is true in order to be able to use the Corresponding Angles Postulate?

2. Draw a diagram representing that information from number 1.

3. What are some other facts you think are true about angle pairs formed by transversals? (Do you think there are more pairs of congruent angles?)

Answers:

1. You need to have parallel lines in order to use the Corresponding Angles Postulate.

2. Create a diagram with two parallel lines and a transversal. (Make sure you put the arrow marks on the parallel lines)

3. Hopefully you can see that the alternate interior angles appear to be congruent as long as the lines are parallel!

We used our answer from question 3 of the warm-up to begin class, by stating a potential theorem: The Alternate Interior Angles Theorem.

From the diagram with parallel lines, it look like the alternate interior angles were congruent.  The key here is that the lines were parallel!  If you look at a transversal crossing two lines that aren’t parallel, the alternate interior angles won’t be congruent.  So, we have conditions for our theorem.

Alternate Interior Angles Theorem

If a transversal crosses two parallel lines, then the alternate interior angles are congruent.

*Note how the theorem is stated very generally.  Recall, to prove it we will create a more specific situation to work with. For example, when we proved the vertical angles theorem, we created angles 1 and 2 to reference specifically, instead of just “two angles”

We will need 3 ideas to be able to complete this proof: The Corresponding Angles Postulate, The Vertical Angles Theorem, and The Transitive Property

Proof:

Given: a // b     Prove: Angle 4 is congruent to angle 5

We also created a diagram to go with this situation:

20140910-155210-57130260.jpg

 

Since a // b, we know that angle 4 is congruent to angle 8 by the corresponding angles postulate.

Since angle 8 and angle 5 are vertical angles, we know that angle 8 is congruent to angle 5 by the vertical angles theorem.

Since angle 8 is congruent to angle 5 and angle 4 is congruent to angle 8, we know that angle 5 is congruent to angle 4 by the transitive property.

The last line shows what we wanted to prove, so we are done!

Notice the different strategies we used: We restated the given information and used that to get new information, we also looked at the diagram to get information, and then for the last line, we combined two pieces of information created in the proof to get even more new information, giving the correct supporting postulates, theorems, and properties along the way.

HW#11 is to try to state and prove the Alternate Exterior Angles Theorem.  Use the same diagram, and a lot of the same language and structure of the proof we just did.  Its really just different angle numbers with a different goal.

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