Problem:
If ∠A=20∘ and ∠AFG=∠AGF, Then ∠B+∠D=
Answer Choices:
A. 48∘
B. 60∘
C. 72∘
D. 80∘
E. 90∘
Solution:
Since ∠AFG=∠AGF and ∠GAF+∠AFG+ ∠AGF=180∘, we have 20∘+2(∠AFG)=180∘. So ∠AFG= 80∘. Also, ∠AFG+∠BFD=190∘, so ∠BFD=100∘. The sum of the angles of △BFD is 180∘, so ∠B+∠D=80∘.
Note: In △AFG,∠AFG=∠B+∠D. In general, an exterior angle of a triangle equals the sum of its remote interior angles. For example, in △GAF,∠x=∠GAF+∠AGF.
Note that, as in Problem 13, some texts use different symbols to represent an angle and its degree measure.
Answer: D.
The problems on this page are the property of the MAA's American Mathematics Competitions
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