Problem:
The ratio of the measures of two acute angles is 5:4, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
Answer Choices:
A. 75
B. 90
C. 135
D. 150
E. 270
Solution:
Let α and β be the measures of the angles, with α<β. Then αβ​=45​. Because α<β, it follows that 90∘−β<90∘−α, so 90∘−α=2(90∘−β). This leads to the system of linear equations 4β−5α=0 and 2β−α=90∘. Solving the system gives α=60∘,β=75∘. The requested sum is α+β=(C)135∘​.
The problems on this page are the property of the MAA's American Mathematics Competitions