Answer :
Final answer:
In a triangle with angle bisectors intersecting at an incenter, if one angle is smaller than the other, then its adjacent angle at the incenter is also smaller. The angle bisectors divide opposite angles proportionally, keeping the relationships between the angles consistent at the incenter.
Explanation:
In triangle ABC with angle bisectors AD, BE, and CF intersecting at incenter I, the statement that if ∠ACB < ∠ABC, then ∠ACD < ∠BEC is true.
This is because the angle bisectors divide the angles proportionally to the sides they are facing. Consequently, if ∠ACB is smaller than ∠ABC, then the angle bisector AD will divide ∠ACB into two angles where each is smaller than the two angles that the angle bisector BE will divide ∠ABC into.
The relationship between the measures of the angles at the incenter and the opposite angles remains consistent, hence ∠AIC is greater than ∠CIB since AD and CF are angle bisectors and the angles at, I am proportional to the angles at A, C, and B respectively.
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