In triangle ABC, AD is the bisector, angle C is 72, and CAD is 13. Find angle B.

Consider triangle ACD:

Since the sum of the angles in the triangle is 180 °, and by the problem statement ∠DCA = 72 °, ∠CAD = 13 °, then:

∠ADC = 180 ° – ∠DCA – ∠CAD = 180 ° – 72 ° – 13 ° = 180 ° – 85 ° = 95 °.

Since AD is the bisector of ∠CAB, then ∠СAD = ∠BAD = 13 °.

Since ∠BDA and ∠ADC are adjacent, their sum is 180 °. Therefore, ∠BDA = 180 ° – ∠ADC = 180 ° – 95 ° = 85 °.

Consider triangle ABD:

Since the sum of the angles in the triangle is 180 °, ∠BAD = 13 °, ∠BDA = 85 °, then ∠ABD = 180 ° – ∠BAD – ∠BDA = 180 ° – 13 ° – 85 ° = 180 ° – 98 ° = 82 °. Therefore, ∠B = 82 °.