2016 AMC 10B Problem 7

1. A ratio of $5:4$ means the angles are $5$ parts and $4$ parts of the same size. Call one part $x$ degrees, so the angles are $5x$ and $4x$.
2. The complement of an angle is what you add to reach $90^\circ$. So the complements are $90 - 5x$ and $90 - 4x$.
3. Which complement is the bigger one? The smaller angle needs more to reach $90$, so $90 - 4x$ is bigger. It must be the one that is twice the other:
$$90 - 4x = 2(90 - 5x).$$
4. Solve: $90 - 4x = 180 - 10x$, so $6x = 90$ and $x = 15$.
5. The angles are $5 \cdot 15 = 75^\circ$ and $4 \cdot 15 = 60^\circ$. Check: their complements are $15^\circ$ and $30^\circ$, and $30$ is twice $15$.
6. The sum is $75 + 60 = 135$.

The answer is $\boxed{\textbf{(C)}\ 135}$.

A ratio like $5:4$ is easiest with one unknown for the size of a "part," which turns the problem into one simple equation. Before writing the equation, decide which quantity is bigger. That choice picks the right equation the first time.

The two angles add to $5x + 4x = 9x$, so the sum must be a multiple of $9$. Among the choices, only $135$ and $270$ are multiples of $9$. Two acute angles add to less than $180$, so the answer is $135$.