Given: ABC is a triangle, BD bisects angle ABC

Prove: AB + BC > AC

angle 3 > angle 1 and angle 4 > angle 2
AB + BC > AD + DC
The exterior angle theorem
Algebra
Given
<1 = <2
Substitution
Definition
BC > DC and AB > AD
If two angles of a triangle are unequal, the sides opposite them are unequal in the same order
Segment addition theorem
AB + BC > AC
Substitution
BD bisects angle ABC
angle 3 > angle 2 and angle 4 > angle 1
AC = AD + DC