AMC 8 · 2009 · #19

Grade 8 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

angle-sum-triangleisosceles-triangle caseworksystematic-enumeration ↑ Prerequisites: angle-sum-triangle

📏 Medium solution 💡 3 insights

Problem

An isosceles triangle has two of its angles measuring 70° and x°. Because we are not told which role each angle plays (base angle or vertex angle), there is more than one possible value of x. Find every possible x and add them up.

Pick an answer.

(A)

(B)

125

(C)

140

(D)

165

(E)

180

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

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.

1STEP 1

Sketch an isosceles triangle: two equal base angles and one vertex angle. Decide which slot each given angle fills.

angles = {a, a, b} with a + a + b = 180°

2STEP 2

Case 1 — both equal angles are 70°, so the vertex x takes what is left of 180°: x = 40.

70 + 70 + x = 180 → x = 180 - 140 = 40

3STEP 3

Case 2 — both equal angles are x° and 70° is the vertex, so 2x + 70 = 180: x = 55.

x + x + 70 = 180 → 2x = 110 → x = 55

4STEP 4

Case 3 — the 70° and x° angles are the equal pair, so x = 70 (the third angle is then 40°, still valid).

x = 70

5STEP 5

All three cases are valid triangles, so the possible x-values 40, 55, and 70 sum to 165.

40 + 55 + 70 = 165 → (D)

Answer

165

Each of x = 40, 55, 70 produces a real triangle: (70,70,40), (55,55,70), (70,70,40) — wait, Case 3's triangle is the same set of angles as Case 1, but x refers to a different angle, so the value of x (70 vs 40) really is different. All three triangles satisfy the 180° sum and have two equal angles, so each x is legitimate. The total 40 + 55 + 70 = 165 matches choice (D), and the answer is below the 180 cap of choice (E), as it should be since no single x can equal 180.

💡Key takeaway

When 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.

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