AMC 10 · 2011 · #7
Grade 8 geometry-2dThe sum of two angles of a triangle is 56 of a right angle, and one of these two angles is 30∘ larger than the other. What is the degree measure of the largest angle in the triangle?
Pick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: The two angles together equal $\frac{6}{5}$ of a right angle ($90^{\circ}$); One of these two angles is $30^{\circ}$ larger than the other; The figure is a triangle, so all three angles add to $180^{\circ}$; Answer choices: (A) $69$, (B) $72$, (C) $90$, (D) $102$, (E) $108$
Unknowns: The degree measure of the largest of the three angles
Understand
Restated: 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.
Givens: The two angles together equal $\frac{6}{5}$ of a right angle ($90^{\circ}$); One of these two angles is $30^{\circ}$ larger than the other; The figure is a triangle, so all three angles add to $180^{\circ}$; Answer choices: (A) $69$, (B) $72$, (C) $90$, (D) $102$, (E) $108$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
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.
Execute — Answer: B
4.NF.B.4 Step 1 Find the sum of the two angles
- A right angle is $90^{\circ}$, and the two angles together are $\frac{6}{5}$ of it.
- Multiply: $\frac{6}{5}\times 90 = 6\times \frac{90}{5} = 6\times 18 = 108$.
- So the two described angles add up to $108^{\circ}$.
💡 Taking a fraction of a right angle is just multiplying the fraction by $90$.
7.EE.B.4 Step 2 Name the angles with a variable
- Let the smaller of the two angles be $x$ degrees.
- The other is $30^{\circ}$ larger, so it is $x+30$.
- Their sum is $108$, which gives one equation: $x + (x+30) = 108$.
💡 One unknown plus a fixed gap captures both angles with a single letter.
7.EE.B.4 Step 3 Solve for the two angles
- Combine like terms: $x + x + 30 = 2x + 30 = 108$.
- Subtract $30$ from both sides: $2x = 78$.
- Divide by $2$: $x = 39$.
- So the smaller angle is $39^{\circ}$ and the larger is $39 + 30 = 69^{\circ}$.
💡 Undo the sum and the difference step by step to split $108$ into two unequal parts.
8.G.A.5 Step 4 Find the third angle
- Every triangle's three angles add to $180^{\circ}$.
- Two of them already use up $108^{\circ}$, so the third angle is what remains: $180 - 108 = 72^{\circ}$.
💡 The leftover of $180^{\circ}$ after the first two angles is the third angle.
4.NBT.A.2 Step 5 Compare and pick the largest
- The three angles are $39^{\circ}$, $69^{\circ}$, and $72^{\circ}$.
- Compare them: $72 > 69 > 39$.
- The largest is $72^{\circ}$, which is the third angle, not either of the two described.
- That matches choice (B).
💡 Line the three measures up and read off the biggest number.
4.NF.B.4 A right angle is $90^{\circ}$, and the two angles together are $\frac{6}{5}$ of 7.EE.B.4 Let the smaller of the two angles be $x$ degrees. The other is $30^{\circ}$ larg 7.EE.B.4 Combine like terms: $x + x + 30 = 2x + 30 = 108$. Subtract $30$ from both sides: 8.G.A.5 Every triangle's three angles add to $180^{\circ}$. Two of them already use up $ 4.NBT.A.2 The three angles are $39^{\circ}$, $69^{\circ}$, and $72^{\circ}$. Compare them: Review
Reasonableness: 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.
Alternative: Skip the variable and use the average. Two numbers summing to $108$ average $54$; since they differ by $30$, they sit $15$ above and below the average, giving $54-15 = 39$ and $54+15 = 69$. Then the third angle is $180 - 108 = 72$, and the largest of $39,\,69,\,72$ is again $72$, confirming (B).
CCSS standards used (min grade 8)
4.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a whole number (Computing $\frac{6}{5}$ of a right angle as $\frac{6}{5}\times 90 = 108^{\circ}$.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Letting the smaller angle be $x$, writing $x + (x+30) = 108$, and solving for $x = 39$.)8.G.A.5Use informal arguments to establish facts about angle sum and exterior angles (Using the triangle angle sum $180^{\circ}$ to find the third angle $180 - 108 = 72$.)4.NBT.A.2Read and write multi-digit whole numbers and compare using symbols (Comparing $39$, $69$, and $72$ to identify the largest angle.)
⭐ 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.
⭐ 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.
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