Skip to main content

2016 AMC 12B Problem 4

Problem 4 of 25EasierAlgebraArithmetic

The ratio of the measures of two acute angles is 5:4,5:4, 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

Show solution

Solution

Let the angles be α<β\alpha\lt\beta with β=54α.\beta=\tfrac54\alpha. The larger complement belongs to the smaller angle, so 90−α=2(90−β)=180−52α.90-\alpha=2(90-\beta)=180-\tfrac52\alpha. This gives 32α=90,\tfrac32\alpha=90, so α=60∘\alpha=60^\circ and β=75∘.\beta=75^\circ. The sum is 135∘.135^\circ. Thus, the correct answer is C.
AoPS wiki

Tagged: ratio and proportion · system of equations

More practice