Competition · AMC preparation · step 4 of 4

AMC 10 · 2012A · #4

Grade 4 geometry-2d
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
optimizationinterval-arithmetic extremal-constructioncasework ↑ Prerequisites: interval-arithmetic
📏 Medium solution 💡 1 insight
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Problem
Rays BA, BC, and BD all start at the same point B. Ray BC makes a 24° angle with ray BA, and ray BD makes a 20° angle with ray BA. What is the smallest possible measure of the angle between rays BC and BD?

Pick an answer.

(A)
0
(B)
2
(C)
4
(D)
6
(E)
12

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The word "smallest" signals a min/max search, so Tool #14 (Extreme Principle) says: list the boundary configurations and pick the extreme one. The only freedom is which side of ray BA ray BD falls on, giving just two cases. Tool #1 (Draw a Diagram) makes those two arrangements visible so the add-versus-subtract choice is obvious.

1STEP 1

Draw the three rays from B

Fix ray BA as the reference. Swing BC out 24° from it; BD is 20° from it — but nothing says which side BD leans.

∠ ABC = 24°, ∠ ABD = 20°
2STEP 2

All angles share ray BA

Both angles are measured from that same ray BA, so BD either sits inside ∠ ABC or on the far side of BA. Only two cases.

3STEP 3

Add or subtract for each case

Same side: BD sits inside ∠ ABC, leaving 24° - 20° = 4°. Opposite sides: the angles stack, 24° + 20° = 44°.

24° - 20° = 4° or 24° + 20° = 44°
4STEP 4

Pick the smaller case

Of 4° and 44°, the smaller is 4°, reached when BD tucks inside ∠ ABC. So the answer is (C).

min(4°, 44°) = 4° → (C)
Answer
4
The two rays are 24° and 20° from a common ray, so on the same side they can differ by no less than 24 - 20 = 4°; they can never coincide, which rules out choice (A) 0. The value 4° sits right at the choices given and matches the difference of the two data, so (C) is consistent. The other extreme, 44°, is the largest possible, confirming 4° as the smallest.
💡Key takeaway

When two rays are measured from the same starting ray, point them the same way to make the angle between them as small as possible — here that leaves just 24 - 20 = 4 degrees.

  • Draw the three rays from B
  • All angles share ray BA
  • Add or subtract for each case
  • Pick the smaller case

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