The Angle Bisector Theorem for Isosceles Triangles

 In an isosceles triangle the bisector of the vertex angle cuts the opposite side in half.

 Note:  The vertex angle of an isosceles triangle is the angle which is opposite a side that might not be congruent to another side.

 To prove this, we rephrase it with a generic isosceles triangle:

If  and  bisects  then .

Proof:

Since  bisects , .   and  share a common side, , and it is given in the hypothesis that .  Therefore  by SAS.  As a consequence,  since these are corresponding sides of the congruent triangles.  So the lengths .  But  by the segment addition postulate, and therefore  (using substitution in algebra).

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