2016 AMC 12B Problem 4

Attempt Problem 4 of the 2016 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

Answer: C

Concepts:ratio and proportion system of equations

Difficulty rating: 1200

Small hint:

Let the angles be

\alpha\lt\beta

with

\dfrac{\beta}{\alpha}=\dfrac54

Big hint:

The complements are

90-\alpha

and

90-\beta;

set

90-\alpha=2(90-\beta)

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.