Question 935779: How do I prove that the angle bisector of the vertex angle of an isosceles triangle is an altitude?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! 
The angle bisector (ray OB) divides the vertex angle AOC into two congruent angles (angle AOB and angle COB).
It also divides the triangle into two triangles (triangle AOB and triangle COB).
You can prove that those two triangles are congruent.
They have one pair of congruent angles (angle AOB and angle COB).
Those angles are flanked by pairs of congruent sides,
because side OA is congruent with side OC (by definition of isosceles triangle),
and side OB is congruent with itself.
So, by SAS congruency, triangle AOB and triangle COB are congruent.
Then, by CPCTC (Congruent Parts of Congruent Triangles are Congruent),
angle ABO and angle CBO are congruent,
but since points A, B, and C are colinear,
angles ABO and CBO being congruent means they are right angles.
Segment BO is the altitude of triangle ACO,
because it is perpendicular to base AC,
and goes from a point on line AC to vertex O.
|
|
|
