AMC 8 · 2017 · #6
Easy mode Grade 8Problem
Picture a triangle. It has three angles inside it.
The three angles are in the ratio 3:3:4. That means if you split the total angle measure into 3+3+4=10 equal parts, the angles take 3 parts, 3 parts, and 4 parts.
The three angles inside any triangle always add up to 180 degrees.
How many degrees is the largest angle?
Pick an answer.
AMC 8 2017 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: A triangle has three interior angles whose measures are in the ratio $3:3:4$. Find the degree measure of the largest of those three angles.
Givens: The three interior angles of a triangle are in the ratio $3:3:4$; The sum of the interior angles of any triangle is $180^\circ$; Answer choices: (A) $18$, (B) $36$, (C) $60$, (D) $72$, (E) $90$ (degrees)
Unknowns: The degree measure of the largest of the three angles
Understand
Restated: A triangle has three interior angles whose measures are in the ratio $3:3:4$. Find the degree measure of the largest of those three angles.
Givens: The three interior angles of a triangle are in the ratio $3:3:4$; The sum of the interior angles of any triangle is $180^\circ$; Answer choices: (A) $18$, (B) $36$, (C) $60$, (D) $72$, (E) $90$ (degrees)
Plan
Primary tool: #7 Identify Subproblems
Secondary: #3 Eliminate Possibilities
Tool #7 (Identify Subproblems) splits the question into two clean pieces: (a) figure out how many degrees one "ratio unit" is worth, then (b) use that unit to size the largest angle. The ratio $3:3:4$ contains $3+3+4=10$ equal units total, and those $10$ units must share the $180^\circ$ of a triangle — so one unit is just $180^\circ \div 10$. Once we know one unit, the largest angle (the $4$-unit part) is immediate. Tool #3 (Eliminate Possibilities) is the back-up check: since this is multiple choice, every candidate answer should be at most $180^\circ$ and should be expressible as $4$ copies of an integer share, which quickly rules out most options.
Execute — Answer: D
3.OA.D.8 Step 1 - Add the parts of the ratio to see how many equal shares the $180^\circ$ gets split into.
- The ratio $3:3:4$ has $3+3+4=10$ parts in total.
💡 Counting up the pieces of a ratio is a one-step addition word problem from Grade 3.
8.G.A.5 Step 2 - Use the fact that the three interior angles of any triangle sum to $180^\circ$.
- So those $10$ equal shares fit exactly inside $180^\circ$, and one share is found by dividing.
💡 Knowing that a triangle's interior angles always add to $180^\circ$ is the Grade 8 informal angle-sum fact.
4.OA.A.2 Step 3 The largest angle is the $4$-share part of the ratio, so multiply one share by $4$ to get its measure in degrees.
💡 "$4$ copies of a share" is exactly the multiplicative-comparison move from Grade 4.
4.MD.C.7 Step 4 - Verify by adding the three angles back together, using the ratio: $3$ shares + $3$ shares + $4$ shares should equal $180^\circ$.
- This rules out every other multiple-choice candidate.
💡 Adding angle parts to check that they make the whole is exactly the Grade 4 "angle measure is additive" idea.
3.OA.D.8 Add the parts of the ratio to see how many equal shares the $180^\circ$ gets spl 8.G.A.5 Use the fact that the three interior angles of any triangle sum to $180^\circ$. 4.OA.A.2 The largest angle is the $4$-share part of the ratio, so multiply one share by $ 4.MD.C.7 Verify by adding the three angles back together, using the ratio: $3$ shares + $ Review
Reasonableness: The three angles work out to $54^\circ, 54^\circ, 72^\circ$ — all positive, all less than $180^\circ$, and summing to exactly $180^\circ$. The triangle is isosceles (two equal small angles) with one slightly larger angle, which matches the $3:3:4$ ratio (the largest is just a bit bigger than the other two, not dramatically so). The largest angle $72^\circ$ is less than $90^\circ$, so this is an acute triangle — consistent with the modest $3:3:4$ ratio. Answer (D) $72^\circ$ is the right size.
Alternative: Tool #6 (Guess and Check) on the answer choices: the largest angle must be exactly $\tfrac{4}{10} = \tfrac{2}{5}$ of $180^\circ$. Quickly compute $\tfrac{2}{5} \times 180 = 72$ — only choice (D) matches. Alternatively check each option by dividing it by $4$ to find a candidate "share" and seeing whether $10$ shares give $180^\circ$: only $72 \div 4 = 18$ and $10 \times 18 = 180$ works.
CCSS standards used (min grade 8)
3.OA.D.8Solve two-step word problems using four operations within 100 (Adding the ratio parts $3+3+4=10$ to count the total number of equal shares.)8.G.A.5Use informal arguments to establish facts about angle sum and exterior angles (Applying the triangle interior angle-sum fact ($180^\circ$) so the $10$ ratio shares can be matched to a known total.)4.OA.A.2Multiply or divide to solve word problems involving multiplicative comparison (Computing the largest angle as $4 \times 18^\circ = 72^\circ$ — "$4$ times as big as one share.")4.MD.C.7Recognize angle measure as additive and solve addition and subtraction problems (Verifying $54^\circ + 54^\circ + 72^\circ = 180^\circ$ by adding the three angle parts to the whole.)
⭐ This AMC 8 problem only needs the Grade 8 fact you already know — the three angles of a triangle always add up to $180^\circ$!
⭐ This AMC 8 problem only needs the Grade 8 fact you already know — the three angles of a triangle always add up to $180^\circ$!
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