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

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 .

2016 AMC 10B Problem 7

Answer: C

Difficulty rating: 1070

Solution:

Problem 7 in Other Years