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## Geometry (all content)

### Course: Geometry (all content)>Unit 12

Lesson 4: Angle bisector theorem

# Intro to angle bisector theorem

The Angle Bisector Theorem states that when an angle in a triangle is split into two equal angles, it divides the opposite side into two parts. The ratio of these parts will be the same as the ratio of the sides next to the angle. Created by Sal Khan.

## Want to join the conversation?

• What does arbitrary mean? Sal uses it when he refers to triangles and angles.
• It just means something random. In this case some triangle he drew that has no particular information given about it.
• I'm a bit confused: the bisector line segment is perpendicular to the bottom line of the triangle, the bisector line segment is equal in length to itself, and the angle that's being bisected is divided into two angles with equal measures. Based on this information, wouldn't the Angle-Side-Angle postulate tell us that any two triangles formed from an angle bisector are congruent? And yet, I know this isn't true in every case. A little help, please?
• Watch out! The bisector is not perpendicular to the bottom line... Imagine you had an isosceles triangle and you took the angle bisector, and you'll see that the two lines are perpendicular. However, if you tilt the base, the bisector won't change so they will not be perpendicular anymore : )
• Hi, instead of going through this entire proof could you not say that line BD is perpendicular to AC, then it creates 90 degree angles in triangle BAD and CAD... with AA postulate, then, both of them are Similar and we prove corresponding sides have the same ratio.
• BD is not necessarily perpendicular to AC. Quoting from Age of Caffiene: "Watch out! The bisector is not [necessarily] perpendicular to the bottom line... Imagine you had an isosceles triangle and you took the angle bisector, and you'll see that the two lines are perpendicular. However, if you tilt the base, the bisector won't change so they will not be perpendicular anymore : ) "
• from to , I have no idea what's going on.
• This video requires knowledge from previous videos/practices. Do the whole unit from the beginning before you attempt these problems so you actually understand what is going on without getting lost :) Good luck!
• If triangle BCF is isosceles, shouldn't triangle ABC be isosceles too?
Here's why:
Segment CF = segment AB. CF is also equal to BC. So BC is congruent to AB. Doesn't that make triangle ABC isosceles?
That can't be right. . . Anybody know where I went wrong?
• Unfortunately the mistake lies in the very first step....
Sal constructs CF parallel to AB not equal to AB.
We know that BD is the angle bisector of angle ABC which means angle ABD = angle CBD. Now, CF is parallel to AB and the transversal is BF. So we get angle ABF = angle BFC ( alternate interior angles are equal). But we already know angle ABD i.e. same as angle ABF = angle CBD which means angle BFC = angle CBD.
Therefore triangle BCF is isosceles while triangle ABC is not.

Hope this helps you and clears your confusion! Best wishes!! :)
• At , Sal says that the two triangles separated from the bisector aren't necessarily similar. This means that side AB can be longer than side BC and vice versa. My question is that for example if side AB is longer than side BC, at wouldn't CF be longer than BC? On the other hand Sal says that triangle BCF is isosceles meaning that the those sides should be the same. I understand that concept, but right now I am kind of confused.
• i think you assumed AB is equal length to FC because it they're parallel, but that's not true. imagine extending A really far from B but still the imaginary yellow line so that ABF remains constant. you can see that AB can get really long while CF and BC remain constant and equal to each other (BCF is isosceles). hope this clears things up
• Sal mentions how there's always a line that is a parallel segment BA and creates the line. Earlier, he also extends segment BD.

How is Sal able to create and extend lines out of nowhere? Is there a mathematical statement permitting us to create any line we want? Can someone link me to a video or website explaining my needs?
• Euclid originally formulated geometry in terms of five axioms, or starting assumptions. The first axiom is that if we have two points, we can join them with a straight line. The second is that if we have a line segment, we can extend it as far as we like.

https://en.wikipedia.org/wiki/Euclidean_geometry#Axioms
• At , what is AA Similarity? I've never heard of it or learned it before....
• If two angles of one triangle are congruent to two angles of a second triangle then the triangles have to be similar. Meaning all corresponding angles are congruent and the corresponding sides are proportional.
• How do I know when to use what proof for what problem? I know what each one does but I don't quite under stand in what context they are used in? this is not related to this video I'm just having a hard time with proofs in general.
• Hi cookid1225!
Great question!
I would suggest that you make sure you are thoroughly well-grounded in all of the theorems, so that you are sure that you know how to use them.
Most of the work in proofs is seeing the triangles and other shapes and using their respective theorems to solve them. It just takes a little bit of work to see all the shapes! You might want to refer to the angle game videos earlier in the geometry course.
For general proofs, this is what I said to someone else:
If you can, circle what you're trying to prove, and keep referring to it as you go through with your proof. Take the givens and use the theorems, and put it all into one steady stream of logic. If you need to you can write it down in complete sentences or reason aloud, working through your proof audibly… If you understand the concept, you should be able to go through with it and use it, but if you don't understand the reasoning behind the concept, it won't make much sense when you're trying to do it.