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2016 AMC 10B Problem 7

Problem 7 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?

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Solution

Let the smaller angle be x.x. Then, the larger angle is 54x.\frac 54 x. Note that their sum is 94x.\frac 94 x. Their complements are then 90−x90-x and 90−54x.90 - \frac 54x. Since 90−x90-x is larger than 90−54x,90-\frac 54 x, we know 90−x=2(90−54x)90−x=180−52xx=60.\begin{aligned}90 -x &= 2\left(90 - \frac 54 x\right) \\90-x&= 180-\frac 52 x\\x&=60.\end{aligned} Therefore, the sum is 94⋅60=135.\dfrac 94 \cdot 60=135. Thus, the correct answer is C .
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