Given: ABC is a triangle, BD bisects ∠ABC

Prove: AB + BC > AC

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