Proof:  Alternate Interior Angles Converse

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.


3 thoughts on “Proof: Alternate Interior Angles Converse

Leave a Reply

Your email address will not be published. Required fields are marked *