2016 AMC 12B Problem 4

Problem 4 of 25EasierAlgebra

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?

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Solution

Let the angles be

\alpha\lt\beta

with

\beta=\tfrac54\alpha.

The larger complement belongs to the smaller angle, so

90-\alpha=2(90-\beta)=180-\tfrac52\alpha.

This gives

\tfrac32\alpha=90,

\alpha=60^\circ

and

\beta=75^\circ.

The sum is

135^\circ.

Thus, the correct answer is C.

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