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2016 AMC 10B Problem 7

2016 AMC 10B Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 2 min · position P1-10

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?

A)
$75$
B)
$90$
C)
$135$
D)
$150$
E)
$270$

Answer

Key insight

Call the angles 5x and 4x; the smaller angle has the larger complement, so 90 - 4x = 2(90 - 5x) gives x = 15 and angles 75 and 60.

Solution

Write the angles as $5x$ and $4x$ degrees. Their complements are $90 - 5x$ and $90 - 4x$ . The smaller angle $4x$ has the larger complement, so the condition must be
$90 - 4x = 2(90 - 5x).$
Expanding, $90 - 4x = 180 - 10x$ , so $6x = 90$ and $x = 15$ .

The angles are $75^\circ$ and $60^\circ$ ; their complements $15^\circ$ and $30^\circ$ do differ by a factor of $2$ , as required. The sum is $75 + 60 = 135$ .

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

Why this works

A ratio $5:4$ is best handled with a single unknown multiplier, turning the problem into one linear equation. Before writing the equation, decide which complement is bigger: the smaller angle has the larger complement. That one observation picks the correct equation and avoids a dead end.

Alternative approach

The sum of the angles is $9x$ , so it must be a multiple of $9$ ; among the choices only $135$ and $270$ qualify, and $270$ would make the angles obtuse. So $135$ .

The trap

Doubling the wrong complement, 90 - 5x = 2(90 - 4x), which yields a negative x and no answer.

Common mistakes

Doubling the wrong complement, $90 - 5x = 2(90 - 4x)$ , which yields a negative $x$ and no answer.
Using supplements ( $180 - \text{angle}$ ) instead of complements.

Techniques

Set up the equation/formula and compute; no special trick needed