Welcome to a lesson on the angle bisector theorem.
The goal of this video is to state and improve the angle bisector theorem.
The angle bisector theorem states that if a point is on the angle bisector
then the point is equidistant from the sides of the angle.
So if this red ray is our angle bisector.
And we consider this point
on the angle bisector.
This theorem tells us that this point would be equidistant.
From the two sides of the angle.
Well in this case the two blue rays.
Meaning this green segment here would be congruent to this green segment here.
Let's see if we can prove this.
Let's first talk about our strategy.
Our strategy is going to be to prove that this triangle here.
Is congruent to this triangle here.
By angle, angle, side.
We'll do this by showing that this angle here.
And this angle here would be
right angles.
And then if this red ray is the angle bisector.
This angle here would be congruent to this angle here.
And then this side of the hypotenuse of the right triangles
would be congruent to itself.
Therefore the two triangles will be congruent by angle, angle, side.
And therefore this segment here is congruent to this segment here.
Let's go ahead and write this out in a two column proof.
So here were given that ray AD.
Bisect angle BAC.
And that ray AB is perpendicular to segment ED.
And
ray AC is perpendicular to segment FD.
So as we normally do we'll start by stating the given.
So if ray AB is perpendicular to segment ED.
We know that angle AED would be a right angle.
And if
ray AC is perpendicular to segment FD.
We know angle AFD would be a right angle.
So angle AED
and
angle
AFD
are right angles.
This would be by definition a perpendicular lines.
And then if these two angles are right angles.
They would be congruent angle AED
is congruent to angle AFD.
And the reason for this is that all right angles are congruent.
So let's go ahead and mark that we know these are right angles.
And that their also congruent.
Now if ray AD bisects angle BAC.
We know that angle
BAD
would be congruent
to angle FAD.
And this is by definition of an angle bisector.
So this angle here is congruent to this angle here.
So we have angle, angle in both of these two triangles.
And now to get the sides notice how both of these triangles share
side AD.
And in both cases it's the hypotenuse of the right triangle.
So for step five we can say that segment AD.
Is congruent to segment AD.
This is by the reflexive property.
So this side is congruent to itself.
So now we know that these two triangles are congruent.
Let's be careful that we state these in corresponding order.
So we say triangle
AED
is congruent
to triangle
AFD.
And the reason for this would be angle, angle, side, congruence.
So if these two triangles are congruent.
Then their corresponding parts must also be congruent.
So segment ED
would be congruent
to segment FD.
And again the reason here would be the corresponding parts.
Of congruent triangles are also congruent.
And this is exactly what we were trying to prove.
If these two segments are congruent.
Of course are the same length.
Which is what the angle bisector theorem tells us.
Any point on this angle bisector
would be equidistant from the two sides
because
from any point on this angle bisector
we would have two congruent triangles.
I hope you found this helpful.