Given: ABC is a triangle, BD bisects ∠ABC

Prove: AB + BC > AC

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