Problem 19 · 2009 AMC 8
Medium
Geometry & Measurement
caseworkisosceles-triangle
Two angles of an isosceles triangle measure 70° and x°. What is the sum of the three possible values of x?
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Answer: D — 165.
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Hint 1 of 2
"Isosceles" only promises SOME two angles are equal — it doesn't say which. That ambiguity is exactly why there are three answers, so don't lock in one picture.
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Hint 2 of 2
Make the equal pair the variable: either 70° matches x°, or the 70° is the repeated angle, or x° is the repeated angle. Use angle-sum 180° in each case.
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Approach: case-split on which angles are the equal pair
- Case 1 — 70° and x° are the equal pair: then x = 70. (Third angle 40°, valid.)
- Case 2 — two 70° angles: the third angle is x = 180 − 70 − 70 = 40.
- Case 3 — two x° angles, with 70° the odd one: 2x + 70 = 180 ⇒ x = 55.
- All three are valid triangles, so sum them: 70 + 40 + 55 = 165.
- Why this transfers: when a shape is "isosceles" but the equal sides/angles aren't specified, systematically try each candidate for the matched pair — and check each case gives a real triangle (positive angles summing to 180°) before counting it.
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