Problem:
The measure of angle ABC is 50∘, AD bisects angle BAC, and DC bisects angle BCA. The measure of angle ADC is
Answer Choices:
A. 90∘
B. 100∘
C. 115∘
D. 122.5∘
E. 125∘
Solution:
Since the sum of the measures of the angles of a triangle is 180∘, in triangle ABC it follows that
∠BAC+∠BCA=180∘−50∘=130∘
The measures of angles DAC and DCA are half that of angles BAC and BCA, respectively, so
∠DAC+∠DCA=2130∘=65∘
In triangle ACD, we have ∠ADC=180∘−65∘=115∘.
Answer: C.
The problems on this page are the property of the MAA's American Mathematics Competitions