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2000 AMC 8

Problem 13

Problem 13 · 2000 AMC 8 Hard
Geometry & Measurement angle-chase
Figure for AMC 8 2000 Problem 13
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Answer: C — 72°.
Show hints
Hint 1 of 2
∠ACT = ∠ATC means triangle CAT is isosceles — those two equal base angles plus the 36° apex must total 180°. That single equation unlocks both base angles at once.
Still stuck? Show hint 2 →
Hint 2 of 2
Once you know ∠ATC, the bisector cuts it in half to give ∠RTC. Then triangle CRT is just another 'three angles sum to 180°' — and notice you already know two of them (∠TCR is the same as ∠ACT).
Show solution
Approach: isosceles to get base angles, then a second triangle's angle sum
  1. Triangle CAT is isosceles (∠ACT = ∠ATC), so 36 + 2·(base angle) = 180, giving each base angle = (180 − 36)/2 = 72°. So ∠ATC = ∠ACT = 72°.
  2. The bisector TR halves ∠ATC, so ∠RTC = 72 ÷ 2 = 36°.
  3. In triangle CRT the angles are ∠TCR = 72° (it's the same corner as ∠ACT), ∠RTC = 36°, and ∠CRT. So ∠CRT = 180 − 72 − 36 = 72°.
  4. The reusable move: an angle chase is just a relay — pin down one angle with isosceles/bisector facts, then carry it into the next triangle's 180° sum. Reuse angles you've already found instead of recomputing them.
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