If alternate interior angles are congruent, then the lines are parallel. Is this statement true or false, and why?

Question

I am taking a geometry class right now and it is pretty difficult. If you would be willing to help me with this, that would be great.

Answer 1

The definition of parallel is that the sum of the angles on each side of the transversal of two lines is two right angles.  From this we can show that the alternate interior angles are congruent. 

In our case, we have that the alternate interior angles are congruent, that is, we have that there are two pairs of alternate interior angles that are congruent, and that together one angle from each pair forms a straight angle, as you go along the lines.  This means that the sum of the angles on each side of the transversal, one from each pair, is two right angles.  The lines are parallel.

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I can't draw here yet.

Answer 2

You should have learned something that sounds a lot like this:

 

Given two parallel lines and a transversal, alternate interior angles are congruent.

 

The key here is that you are told the two lines are parallel, and you have learned that the relationships between the angles created by the intersection of the transversal line and the two parallel lines.

 

alternate interior angles are congruent

alternate exterior angles are congruent

corresponding angles are congruent

same side interior angles are supplementary

same side exterior angles are supplementary

 

All of the above are true based on the two lines being parallel with a transversal line intersecting.

 

This question asks if the reverse is true.

 

If you are told that there are three lines intersecting in a way to form alternate interior angles, and if those alternate interior angles are congruent, will that force two of the lines to be parallel?

 

If you can think of even one way to form congruent alternate interior angles without the two lines being parallel, then it is false. If there is no way to do that, then it is true.

 

Now think about it and write your answer.