2011 AMC 10B Problem 7

P1-10~2 minPrint

Angles & Polygons Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

The two angles add to 108 degrees, so they are 39 and 69. The third angle is 180 minus 108, which is 72, and it is the biggest.

Solution

A right angle is $90^\circ$ . So the two angles add up to $\frac{6}{5} \times 90 = 108$ degrees.
One of them is $30^\circ$ more than the other. Take away the extra $30$ : $108 - 30 = 78$ , which splits evenly as $39 + 39$ . So the angles are $39^\circ$ and $39 + 30 = 69^\circ$ .
The question asks about the whole triangle, so we also need the third angle. The angles of a triangle add to $180^\circ$ , so the third one is $180 - 108 = 72$ degrees.
The three angles are $39^\circ$ , $69^\circ$ and $72^\circ$ . The largest is $72^\circ$ .

The answer is $\boxed{\textbf{(B)}\ 72}$ .

Why this works

The problem tells you about only two of the angles, but asks about the largest angle of the whole triangle. Always find all three angles (using the $180^\circ$ total) before choosing the largest or smallest.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(B)}\ 72}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 2: Angles
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Geometry, Chapter 9: Polygons (§9.1, §9.2)
Introduction to Algebra, Chapter 7: Proportion (§7.4)