2016 AMC 10B Problem 7

Question

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 linear equation

Difficulty rating: 1070

Small hint:

Let the smaller acute angle be

x

Big hint:

The larger angle has the smaller complement

Solution:

Let the smaller angle be

x.

Then, the larger angle is

\frac 54 x.

Note that their sum is

\frac 94 x.

Their complements are then

90-x

and

90 - \frac 54x.

Since

90-x

is larger than

90-\frac 54 x,

we know

\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

\dfrac 94 \cdot 60=135.

Thus, the correct answer is C .