Given: ABC is a triangle, BD bisects ∠ABC

Prove: AB + BC > AC

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