Given: ABC is a triangle, BD bisects ∠ABC

Prove: AB + BC > AC

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