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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!! :)
• I hope you got that water
• 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
• That makes no sense
• 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?