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Vertex angle
They're the 'base angles'.
Not necessarily. The only time that the angle bisector would bisect the opposite side is if you were bisecting the vertex angle of an isosceles triangle.
A median of a triangle is a line or segment that passes through a vertex and the midpoint of the side opposite that vertex. The median only bisects the vertex angle from which it is drawn when it is an isosceles triangle.
If you mean the vertex where the two equal sides of an isosceles triangle intersect, the side is the base.