2012 AMC 10A Problem 4

Both angles have vertex $B$ and share ray $BA$. Ray $BC$ makes a $24^\circ$ angle with ray $BA$, and ray $BD$ makes a $20^\circ$ angle with it. There are two possible pictures.

If $C$ and $D$ are on opposite sides of line $BA$, the angles sit next to each other and $\angle CBD = 24^\circ + 20^\circ = 44^\circ$.

If $C$ and $D$ are on the same side of ray $BA$, the angle $ABD$ lies inside angle $ABC$, and $\angle CBD = 24^\circ - 20^\circ = 4^\circ$.

The smaller of the two is $4^\circ$; it cannot be $0$ because the two rays make different angles with $BA$ and so cannot coincide.

The answer is $\boxed{\textbf{(C)}\ 4}$.

Two angles that share a vertex and one ray combine in exactly two ways: they add (opposite sides) or subtract (same side). "Smallest possible" is a hint that the configuration is not fixed and you should compare both options.

• Assuming the rays are on opposite sides and answering 44, or assuming the angles cannot overlap at all.
• Choosing $0$ by imagining rays $BC$ and $BD$ coinciding, which is impossible since $24^\circ \neq 20^\circ$.