2016 AMC 12B Problem 4

Below is the professionally curated solution for Problem 4 of the 2016 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12B solutions, or check the answer key.

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Concepts:ratio and proportion system of equations

Difficulty rating: 1200

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?

$75$

$90$

$135$

$150$

$270$

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 $\tfrac32\alpha=90,$ so $\alpha=60^\circ$ and $\beta=75^\circ.$ The sum is $135^\circ.$

Thus, the correct answer is C.

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