SOLUTION: Conjecture: The angle bisector of the vertex angle of an isosceles triangle is also a median to the base.

Question 990717: Conjecture: The angle bisector of the vertex angle of an isosceles triangle is also a median to the base.
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!

The angle bisector divides it into two triangles which can be proved congruent by SAS. One pair of congruent sides are the congruent legs of the isosceles triangle. The angles are congruent because an angle bisector divides the angle into two congruent parts. The other pair of congruent sides is the angle bisector itself, as it is part of both triangles. So you have SAS. That makes the angle bisector also a bisector of the base because of corresponding parts of conruent triangles. That is all that is needed to prove it is a median. ----------------------------------------------------- It is also the perpendicular bisector of the base, which you may also be asked to prove later. To do that, you prove additionally that the angles that the angle bisector make with the base are right angles. First, they are congruent because of corresponding parts of congruent triangles. Also they form a linear pair. Then congruent angles that form a linear pair are right angles. Edwin

Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
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See the lesson An altitude a median and an angle bisector in the isosceles triangle in this site.

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