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2011 AMC 10B Problem 7

2011 AMC 10B Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Angles & Polygons · difficulty 2 of 5 · about 2 min · position P1-10

The sum of two angles of a triangle is $6/5$ of a right angle, and one of these two angles is $30^{\circ}$ larger than the other. What is the degree measure of the largest angle in the triangle?

A)
$69$
B)
$72$
C)
$90$
D)
$102$
E)
$108$

Answer

Key insight

The two angles total 108 degrees, so they are 39 and 69, and the third angle 180 - 108 = 72 is the largest.

Solution

Two of the angles sum to $\frac{6}{5} \cdot 90^\circ = 108^\circ$ . If they are $x$ and $x + 30$ , then $2x + 30 = 108$ , so $x = 39$ and the two angles are $39^\circ$ and $69^\circ$ .

The third angle is $180^\circ - 108^\circ = 72^\circ$ .

Comparing $39, 69, 72$ , the largest angle is $\boxed{\textbf{(B)}\ 72}$ .

Why this works

The problem describes two angles but asks about the whole triangle, so the angle-sum $180^\circ$ must be used to find the third one. Always list all three angles before answering a "largest" or "smallest" question.

The trap

Reporting 69, the larger of the two described angles, without computing the third angle.

Common mistakes

Reporting $69$ , the larger of the two described angles, without computing the third angle.
Reading " $6/5$ of a right angle" as $108$ being one angle rather than the sum of two, which leads to choice (E).

Techniques

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