is the new angle exactly half the original apex angle, for a triangle cut in half through that apex? Let each apex (point) of a triangle be apex $A$, apex $B$, and apex $C$. Let each length then be $\overline {AB},\overline {AC},\overline {BC}$. If triangle is cut in half, through an imaginary line from apex $A$ down to the exact middle (bisection) of $BC$, then will the new angle at apex $A'$ (angle of apex $C$ to apex $A$ to the point at $\frac{\overline {BC}}2$) be exactly equal to half the old angle at apex $A$ (that is equal to angle $\frac {B\hat AC}2$)? That is, does new angle $A'=\frac A2$ ?

Ratio of sides and cut parts of opposite side is same for angle bisection at A. $ AB/AC = AM/MC $ So, unless the triangle is isosceles ( AB= AC) it will not be so.