AMC 10 · 2012 · #4
Grade 4 geometry-2dLet ∠ABC=24∘ and ∠ABD=20∘. What is the smallest possible degree measure for ∠CBD?
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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