AMC 10 · 2012 · #4
Easy mode Grade 4Three rays all start at the same point B. Ray BC makes a 24∘ angle with ray BA. Ray BD makes a 20∘ angle with ray BA, but it can lean to either side. What is the smallest the angle between ray BC and ray BD can be?
Pick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Ray $BC$ makes a $24^\circ$ angle with ray $BA$, and ray $BD$ makes a $20^\circ$ angle with ray $BA$. All three rays start at the same point $B$. Find the smallest the angle between $BC$ and $BD$ can be.
Givens: $\angle ABC = 24^\circ$; $\angle ABD = 20^\circ$; Rays $BA$, $BC$, $BD$ all share the vertex $B$; Answer choices: (A) $0$, (B) $2$, (C) $4$, (D) $6$, (E) $12$
Unknowns: The smallest possible value of $\angle CBD$, the angle between rays $BC$ and $BD$
Understand
Restated: Ray $BC$ makes a $24^\circ$ angle with ray $BA$, and ray $BD$ makes a $20^\circ$ angle with ray $BA$. All three rays start at the same point $B$. Find the smallest the angle between $BC$ and $BD$ can be.
Givens: $\angle ABC = 24^\circ$; $\angle ABD = 20^\circ$; Rays $BA$, $BC$, $BD$ all share the vertex $B$; Answer choices: (A) $0$, (B) $2$, (C) $4$, (D) $6$, (E) $12$
Plan
Primary tool: #14 Extreme Principle
Secondary: #1 Draw a Diagram
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.
Execute — Answer: C
4.G.A.1 Step 1 Draw the three rays from B
- Start with ray $BA$ as a reference.
- Swing ray $BC$ out $24^\circ$ from it.
- Ray $BD$ is $20^\circ$ from $BA$, but nothing says on which side, so it could point $20^\circ$ away toward $C$'s side, or $20^\circ$ the other way.
💡 Fixing one ray and drawing the others against it turns three loose facts into one picture you can measure.
4.MD.C.5 Step 2 All angles share ray BA
- Both given angles are measured from the same ray $BA$.
- So ray $BD$ sits either between $BA$ and $BC$ (same side), or on the far side of $BA$ away from $BC$.
- Those are the only two positions.
💡 When two angles are measured from a shared ray, their overlap depends only on which side the second ray leans.
4.MD.C.7 Step 3 Add or subtract for each case
- Same side: ray $BD$ lies inside $\angle ABC$, so $\angle CBD$ is what's left after removing $\angle ABD$: $24^\circ - 20^\circ = 4^\circ$.
- Opposite sides: the two angles stack up around ray $BA$: $24^\circ + 20^\circ = 44^\circ$.
💡 Angle measures add when they don't overlap and subtract when one sits inside the other.
4.MD.C.7 Step 4 Pick the smaller case
- The question wants the smallest $\angle CBD$.
- Between $4^\circ$ and $44^\circ$, the smaller is $4^\circ$, which happens when ray $BD$ tucks inside $\angle ABC$.
- So the answer is (C).
💡 To make the gap between two rays smallest, aim them as much the same direction as the rules allow.
4.G.A.1 Start with ray $BA$ as a reference. Swing ray $BC$ out $24^\circ$ from it. Ray $ 4.MD.C.5 Both given angles are measured from the same ray $BA$. So ray $BD$ sits either b 4.MD.C.7 Same side: ray $BD$ lies inside $\angle ABC$, so $\angle CBD$ is what's left aft 4.MD.C.7 The question wants the smallest $\angle CBD$. Between $4^\circ$ and $44^\circ$, Review
Reasonableness: The two rays are $24^\circ$ and $20^\circ$ from a common ray, so on the same side they can differ by no less than $24 - 20 = 4^\circ$; they can never coincide, which rules out choice (A) $0$. The value $4^\circ$ sits right at the choices given and matches the difference of the two data, so (C) is consistent. The other extreme, $44^\circ$, is the largest possible, confirming $4^\circ$ as the smallest.
Alternative: Use a number line of directions from ray $BA$: $BC$ points to $+24^\circ$, and $BD$ points to either $+20^\circ$ or $-20^\circ$. The gap $\angle CBD$ is $|24 - 20| = 4^\circ$ or $|24 - (-20)| = 44^\circ$. The absolute difference is smallest when both directions carry the same sign, again giving $4^\circ$ and (C).
CCSS standards used (min grade 4)
4.G.A.1Draw points, lines, line segments, rays, angles, and identify in figures (Drawing rays $BC$ and $BD$ at their given angles from ray $BA$ to picture the two arrangements.)4.MD.C.5Recognize angles as geometric shapes formed when two rays share an endpoint (Seeing that both $\angle ABC$ and $\angle ABD$ are measured from the shared ray $BA$, which sets up the two cases.)4.MD.C.7Recognize angle measure as additive and solve addition and subtraction problems (Computing $24^\circ - 20^\circ = 4^\circ$ (same side) and $24^\circ + 20^\circ = 44^\circ$ (opposite sides), then taking the smaller.)
⭐ 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.
⭐ 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.
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