Angle Bisector Theorem
Warning: This name is used differently in different textbooks. In some textbooks, it refers to the theorem which states that any point on an angle bisector is equidistant from the two sides of the angle.
What most textbooks call the Angle Bisector Theorem is this:
An angle bisector in a triangle divides the opposite side into two segments which are in the same proportion as the other two sides of the triangle.
In the figure above, bisects , so .
Triangle Angle Bisector Theorem
An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.
By the Angle Bisector Theorem,
Extend to meet at point .
By the Side-Splitter Theorem,
The angles and are corresponding angles.
Since is a angle bisector of the angle , .
By the Alternate Interior Angle Theorem , .
Therefore, by transitive property, .
Since the angles and are congruent , the triangle is an isosceles triangle with .
Replacing by in equation (1),
Find the value of .
By Triangle-Angle-Bisector Theorem,
Divide both sides by .
The value of is .