# Proof: Alternate Interior Angles Converse

Welcome to a Proof of

the Alternate Interior Angles Converse. This theorem states that

is two lines are cut by a transversal and the

alternate interior angles are congruent, then

the lines are parallel. So we want to be able to

prove that if angle three is congruent to angle six,

or angle four is congruent to angle five, we have parallel lines. And the prove of this is

going to be based upon the Corresponding Angles

Converse Postulate, which states that if two

lines are cut by a transversal and the corresponding

angles are congruent, then the lines are parallel. So we’re going to have

to end up showing that the corresponding angles

are congruent, and therefore by this postulate, the

lines will be parallel. Let’s take a look at our proof. We’re given that transversal

t cuts line l and m, and we know that angle three

is congruent to angle six. So this angle here is

congruent to this angle here, and those are alternate interior angles. So before we start, we want

to devise a strategy so that we can get corresponding

angles to be congruent, and therefore the lines will be parallel. Notice that angle two and angle

six are corresponding angles and angle two happens

to be a vertical angle with angle three. By using the definition of

vertical angles, we can say that angle three and angle two are

congruent, and that angle two is congruent to angle six,

and therefore the lines will be parallel. So let’s go ahead and get started. Again, the first step is almost always to state the given information. So we know that line t cuts lines l and m with angle three and

angle six being congruent. And again ultimate goal

is to show that angle two and angle six are congruent. So the first thing I’m going to

do is restate this congruence, I’m going to state that angle

six is congruent to angle three, and this is by the symmetric property. And the reason I’m changing

the order here is because of the transitive property which

we’ll see in just a minute. So for number three we’re going

to go ahead and state that angle three is congruent to angle two, because vertical angles are congruent. I could also say by definition

a vertical angle or something similar to that. Now you can probably see why

I use a symmetric property in step two. If we have angle six

congruent to angle three, and angle three congruent to angle two, we can use the transitive

property to state that angle six would be congruent to angle two. In order to apply the transitive

property we have to have these congruent angles

linked, meaning angle six is congruent to angle three,

and angle three is congruent to angle two so they connect. So again the justification here,

is the transitive property. And if we know that angle six

and angle two are congruent, these are corresponding

angles and therefore the lines are parallel from

the corresponding angles converse postulate. And this is one possible proof

that if we have a transversal that cuts two lines, and the alternate interior

angles are congruent, the two lines will be parallel. I hope you found this helpful.

Hello 6th hour geometry 😀

Bless you

Vertical angles are not by definition congruent. You should state that as a theorem.