Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #19
Grade 8 geometry-2dPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The phrase "sum of the three possible values" tells us there are exactly three cases — a perfect cue for Tool #2 (Systematic List). We organize the cases by asking which two of the three angles are the equal pair: either the two 70° angles are equal, the two x° angles are equal, or the 70° and x° are the equal pair. Tool #1 (Draw a Diagram) supports this — a quick sketch with the equal sides marked makes each case concrete and prevents double-counting.
Label the isosceles angles
Sketch an isosceles triangle: two equal base angles and one vertex angle. Decide which slot each given angle fills.
Recognizing the isosceles structure — two equal angles plus one other — is the Grade 4 "classify triangles by their properties" skill.
4.G.A.2Draw A DiagramTry 70 as the base angles
Case 1 — both equal angles are 70°, so the vertex x takes what is left of 180°: x = 40.
The triangle-angle-sum fact (180°) is the Grade 8 "informal arguments about triangle angles" standard.
8.G.A.5Make A Systematic ListTry x as the base angles
Case 2 — both equal angles are x° and 70° is the vertex, so 2x + 70 = 180: x = 55.
Same angle-sum equation, different slot for the unknown — a clean Tool #2 sub-case.
When the given 70° angle is the vertex angle, each of the two equal base angles measures 55°.
▸ Why?
The two base angles rest against the two equal sides, so they must be equal to each other; call each one x°.
▸ Why?
Folding the triangle along the line through the vertex lays one equal side onto the other and carries one base angle exactly onto the other, so their measures match.
▸ Why?
Those two equal base angles together with the 70° vertex angle are the triangle's three angles, so x + x + 70 = 180.
▸ Why?
The three corners of any flat triangle always fill exactly one straight angle.
▸ Why?
Undoing 2x + 70 = 180 — take the 70 back off, then share the remaining 110 equally between the two angles — forces each base angle to one value.
▸ Why?
Subtraction reverses the added 70 and division reverses the doubling, so every step is forced and leaves a single answer.
Pair 70 with x
Case 3 — the 70° and x° angles are the equal pair, so x = 70 (the third angle is then 40°, still valid).
By definition, the two base angles of an isosceles triangle have the same measure — this is the Grade 4 classification fact.
4.G.A.2Make A Systematic ListAdd the three values
All three cases are valid triangles, so the possible x-values 40, 55, and 70 sum to 165.
Adding the three possible angle measures uses the Grade 4 "angles are additive" skill in its simplest form.
4.MD.C.7Make A Systematic ListWhen a problem says "the possible values," make a short list of every case — here, just three ways to label the equal angles — and the answer falls out by adding.
- Label the isosceles angles
- Try 70 as the base angles
- Try x as the base angles
- Pair 70 with x
- Add the three values
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