AMC 10 · 2011 · #7

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

angle-sum-trianglesystems-of-equationslinear-equations-one-var convert-to-algebra ↑ Prerequisites: angle-sum-triangle

📏 Medium solution 💡 2 insights

Problem

In a triangle, two of the angles add up to

\frac{6}{5}

of a right angle, and one of those two is

30^\circ

bigger than the other. Find the degree measure of the triangle's largest angle.

Pick an answer.

(A)

(B)

(C)

(D)

102

(E)

108

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Two unknown angles are tied together by a sum and a difference, which is exactly the setup Tool #4 (Introduce a Variable) handles: name the smaller angle, write the larger as that plus 30, and turn the words into one equation. Tool #7 (Identify Subproblems) splits the work into clean pieces — first the sum of the two angles, then each angle, then the third angle from the triangle. Tool #3 (Eliminate Possibilities) finishes it: compare all three angle measures to pick the true largest, since the biggest one may be the leftover third angle rather than either named angle.

1STEP 1

Find the sum of the two angles

A right angle is $90^\circ$ and the two given angles are $\frac{6}{5}$ of it, so together they make $\frac{6}{5}\times 90 =$ $108^\circ$ .

6/5 × 90^° = 108^°

2STEP 2

Name the angles with a variable

Let the smaller angle be $x$ ; the other is $30^\circ$ bigger, so it is $x+30$ . Their sum gives one equation: $x+(x+30)=108$ .

x + (x+30) = 108

3STEP 3

Solve for the two angles

Combine and undo: $2x+30=108$ , so $2x=78$ and $x=39$ — the two angles are $39^\circ$ and $69^\circ$ .

2x + 30 = 108 → 2x = 78 → x = 39, x+30 = 69

4STEP 4

Find the third angle

A triangle's three angles total $180^\circ$ , and two of them already use $108^\circ$ , so the third is $72^\circ$ .

180 - 108 = 72^°

5STEP 5

Compare and pick the largest

The angles are $39^\circ$ , $69^\circ$ , $72^\circ$ , so the largest is $72^\circ$ — the third angle, not either named one. Choice (B).

max(39, 69, 72) = 72 → (B)

Answer

Add all three angles back: 39 + 69 + 72 = 180, exactly a triangle's total, so the split is consistent. The two named angles 39 and 69 differ by 30 and sum to 108, matching both clues. The largest, 72, is bigger than the tempting 69 (choice A, the larger named angle) and smaller than 108 (choice E, the whole sum) — so the trap answers are the two named-angle numbers, and 72 sits sensibly between them as the third angle.

💡Key takeaway

Turn the two clues into one equation to get both angles, use the $180^\circ$ triangle rule for the third, then compare all three to find the biggest.

Find the sum of the two angles
Name the angles with a variable
Solve for the two angles
Find the third angle
Compare and pick the largest

Find the sum of the two angles

Name the angles with a variable

Solve for the two angles

Find the third angle

Compare and pick the largest

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