Final answer:
The median from the vertex of an isosceles triangle is proven to be the angle bisector of the vertex angle by showing that the triangle has two equal sides which create two congruent triangles when a median is drawn, thus bisecting the vertex angle.
Step-by-step explanation:
To prove that the median from the vertex of an isosceles triangle is also the angle bisector of the vertex angle, we first recall that an isosceles triangle has two sides of equal length, and the angles opposite those sides are also equal. Let's consider an isosceles triangle ABC, with AB = AC.
Let D be the midpoint of base BC. Because AD is a median, BD = CD. Draw segment AD. Triangle ABD and triangle ACD are formed and because AB = AC, and BD = CD (by construction), and AD is shared by both triangles, we can say that triangles ABD and ACD are congruent by the Side-Side-Side postulate (SSS).
Due to this congruence, we know that the corresponding angles are equal. Hence, angle BAD is equal to angle CAD. This means that the median AD not only bisects the base BC but also bisects the vertex angle BAC, hence proving AD is an angle bisector as well as a median.