Welcome to a video of the proof of the sum of the interior angles of a triangle
is one hundred eighty degrees. This is also called the triangular sum theorem.
This proof it's going to be based upon having two parallel lines
with triangle ABC as we see here. So let's talk about our strategy before we write
the proof.
Since line AD is parallel to line BC,
and we have two transversals, angle one and angle four would be
alternate interior angles, and therefore they would be congruent.
So the measure of angle one is equal to the measure of angle four,
and the measure of angle five would be equal to the measure of angle two.
Notice angles four, three and five
together form a straight angle that would have a measure of one hundred eighty degrees.
So if we can show the sum of these three angles equal up to one hundred eighty degrees,
and we know the measure of angle four is equal to the measure of angle one,
and the measure of angle five is equal to the measure of angle two, we can perform a
substitution to show
the sum of the measures of angle one, two, and three,
will be equal to one hundred eighty degrees. Now there are a couple of other minor details we want to
address,
but that's the main idea behind the proof, so let's go ahead and get started.
So again, here we have the same diagram as on the previous screen, just a little bit
smaller.
So of course we'll start with the given, we have a triangle ABC
with line AD parallel to line BC.
Next we'll state angle four is congruent to angle one, and angle five is congruent to angle two,
and the reason would be the alternate interior angles theorem.
And since these angles are congruent, we know their measures would be equal.
So the measure of angle four is equal to the measure of angle one,
and the measure of angle five is equal to the measure of angle two.
This is by
definition of congruent angles.
Now in order to show that the sum
of the measure of angles four, three and five
is one hundred eighty degrees, we going to have to use the linear pair postulate.
So now we're going to say that the measure of angle four
plus the measure of angle
BAD,
notice angles three and five form angle BAD,
is equal to one hundred eighty degrees,
and this is by the linear pair postulate.
Now our ultimate goal is to replace the measure of BAD
with the measure of angle three, plus the measure of angle five.
We'll first have to state that the measure of angle three, plus the measure of angle five
is equal to the measure of angle BAD.
And the reason for this is the angle addition postulate.
Now we can replace the measure of angle BAD with the measure of angles three, plus
the measure of angle five
using substitution. So the measure of angle four
plus the measure angle three, plus the measure of angle five
is equal to one hundred eighty degrees.
And the reason is the substitution property.
Now I think we have it. Go go back up to step three, notice at the measure of angle four is equal
to the measure of angle one.
So by performing substitution we'd have the measure of angle one,
plus the measure of angle three, and then we can replace the measure of angle five
with the measure of angle two,
and this is what we're trying to prove.
And the justification for this last step was the same as
the reason for number six, the substitution property.
And that's going to do for this video. I hope you found it helpful.