Question

What do you need to do to prove a line is an angle bisector of an angle?

Answer 1

Answer: The answer is given below.

Step-by-step explanation:We are asked the method to prove that a line is an angle bisector of an angle.

The angle bisector theorem states that an angle bisector of an angle divides the opposite side of a triangle into two segments that are proportional to the triangle's other two sides.

Consider the attached figure, where we need to prove that AD is the angle bisector of ∠BAC.

Therefore, according to the angle bisector theorem, if we can show

$\dfrac{AB}{BD}=\dfrac{AC}{CD}$, then we can conclude that AD is the angle bisector of ∠BAC.

Hence explained.

Answer 2

Using a coordinate geometry approach, identify the coordinates of the vertex of the angle and the equations of the lines forming the two sides; choose an arbitrary point on each line and find the general equation of the line connecting them (the third side of your triangle); write the equation of the line that meets the conditions of angle bisection (that it is equidistant from each of the lines forming the two sides); solve simultaneously the equations for this line and for the third side.

If you are trying to do this as an absolute proof for any angle and triangle, your equations will be full of unknowns (x1, y1, m1, etc), and will need a lot of careful algebraic manipulation. If you have a specific triangle in mind, the presence of numbers makes the solution of the equations much simpler.

Of course, this is not the only method of proof available, but it is the simplest to describe as a general procedure without actually writing out the required proof!

More intuitively, since the angle bisector must be midway between the two rays that form the adjacent sides of the triangle, it must cross any line which intersects those two rays, which the third side of the triangle must do. This is very hard to show as a proof without using diagrams.