AMC 8 · 2017 · #6
Grade 8 geometry-2drate-ratioPick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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° of a triangle — so one unit is just 180° ÷ 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° and should be expressible as 4 copies of an integer share, which quickly rules out most options.
Add the ratio parts: 3 + 3 + 4 = 10 equal shares split the 180°.
Counting up the pieces of a ratio is a one-step addition word problem from Grade 3.
3.OA.D.8Identify SubproblemsAny triangle's three angles sum to 180°, so one share is 180° ÷ 10 = 18°.
Knowing that a triangle's interior angles always add to 180° is the Grade 8 informal angle-sum fact.
8.G.A.5Identify SubproblemsThe largest angle is the 4-share part, so 4 × 18° = 72°.
"4 copies of a share" is exactly the multiplicative-comparison move from Grade 4.
4.OA.A.2Identify SubproblemsCheck: 54° + 54° + 72° = 180°, which rules out every other choice.
Adding angle parts to check that they make the whole is exactly the Grade 4 "angle measure is additive" idea.
4.MD.C.7Eliminate PossibilitiesThis AMC 8 problem only needs the Grade 8 fact you already know — the three angles of a triangle always add up to 180°!