Welcome to a proof of a theorem if one side of a triangle
is longer than another
then the angle opposite the longer side
is greater than the angle opposite the shorter side.
So if we had a triangle like this
notice the side opposite angle C is 21 units.
And this side
opposite angle B is 12 units.
Therefore by this theorem
the measure of angle C
would be greater than
the measure of angle B.
And since the side opposite angle C is actually the longest side of the triangle.
We could also conclude that the measure of angle C
is greater than the measure of angle A.
Notice this side opposite angle A is only 15 units.
Let's see if we can prove this theorem.
So were given that the length of AC
is greater than the length of AB.
We want to prove that the measure of angle B is greater than the measure of angle C.
This is kind of a tricky proof.
What we're going to have to do here is
pick a point x
on side AC
to make
triangle
ABX an isosceles triangle.
So let's get started on that idea.
Number one we'll state the given that the length of AC is greater than
the length of AB.
And now we're gonna locate this point x
Somewhere in here.
Such that the length of AB will be equal to the length of AX.
Therefore we're going to have an isosceles triangle
on the right side.
So for step two
we'll locate point x.
The length of AB
will be equal to the length of AX.
And the reason for this is the ruler postulate.
So now we know that
triangle ABX is an isosceles triangle.
And the reason for this is just the definition
of an isosceles triangle.
Remember if we have an isosceles triangle.
The base angles are congruent.
And therefore equal in measure.
So to keep track of all these angles. Let's go ahead and number them.
Let's call this angle one. We'll call this angle two.
And we'll call this angle three.
Again because triangle ABX is isosceles.
We know the measure of angle one is equal to the measure of angle two.
This is by the base angles theorem.
And now we're gonna create some equations
that involves angle one, two, three
B and C.
The first equation is going to involve angle two
as an exterior angle.
Of triangle BCX.
So if angle two is an exterior angle
the measure of angle two
would be equal to
the sum of the two non adjacent angles
or the measure of angle three
plus the measure of angle C.
And the reason for this is the exterior
angle theorem.
Now using statement four and statement five.
We can perform substitution
for the measure of angle two.
We can say the measure of angle one is equal to
the measure of angle three.
Plus the measure of angle C.
Again this is by the substitution property.
Now let's take a look at angle B.
Now it's also true the measure of angle B
will be equal to the measure of angle three plus the measure of angle one
based upon how we set this up.
And this would be true by the
angle addition postulate.
And now we want to be able to create an equation here.
that involves both
the measure of angle B
and
the measure of angle C.
And we can do that now because we can perform a substitution
for the measure of angle one.
For statement eight
we'll have the measure of angle B
equals
the measure of angle three plus
we'll replace
the measure of angle one with the measure of angle three
plus the measure of angle C.
And again this is by the substitution property.
I think now we have enough information to prove
that measure angle B is greater than the measure angle C.
Because in order for these to be equal
notice how we have to add
the measure angle three twice
to the measure of angle C.
So the measure of angle B must be greater than the measure of angle C.
And this would be by definition of greater than.
I hope the flow of this prove make sense. And I hope you found it helpful.