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 Ξ± and Ξ² be the measures of the angles, with Ξ±<Ξ². Then Ξ±Ξ²β=45β. Because Ξ±<Ξ², it follows that 90ββΞ²<90ββΞ±, so 90ββΞ±=2(90ββΞ²). This leads to the system of linear equations 4Ξ²β5Ξ±=0 and 2Ξ²βΞ±=90β. Solving the system gives Ξ±=60β,Ξ²=75β. The requested sum is Ξ±+Ξ²=135ββ.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions