measure of arc BD

The angle formed by two secants intersecting outside a circle is half the difference of the intercepted arcs. Let the measure of arc BD be $x$. Then $30^\circ = \frac{1}{2} (m(\text{arc AE}) - m(\text{arc BD}))$. Given $m(\text{arc AE}) = 96^\circ$, we have $30^\circ = \frac{1}{2} (96^\circ - x)$. Solving for $x$: $60^\circ = 96^\circ - x$, so $x = 96^\circ - 60^\circ = 36^\circ$. However, looking at the diagram, angle C intercepts arcs BD and AE. The formula is $m\angle C = \frac{1}{2} |m(\text{arc AE}) - m(\text{arc BD})|$. So, $30^\circ = \frac{1}{2} |96^\circ - m(\text{arc BD})|$. This gives $60^\circ = |96^\circ - m(\text{arc BD})|$. This implies either $96^\circ - m(\text{arc BD}) = 60^\circ$ or $96^\circ - m(\text{arc BD}) = -60^\circ$. Case 1: $m(\text{arc BD}) = 96^\circ - 60^\circ = 36^\circ$. Case 2: $m(\text{arc BD}) = 96^\circ + 60^\circ = 156^\circ$. From the diagram, arc BD is clearly smaller than arc AE, so $36^\circ$ is the correct value. Let's re-examine the diagram and the options. The options are 66, 33, 63, 36. My calculation yields 36. Let's check if there's another interpretation. The angle $30^\circ$ is $\angle C$. The intercepted arcs are arc AE and arc BD. The formula is $m\angle C = \frac{1}{2} (m(\text{arc AE}) - m(\text{arc BD}))$. So $30^\circ = \frac{1}{2} (96^\circ - m(\text{arc BD}))$. $60^\circ = 96^\circ - m(\text{arc BD})$. $m(\text{arc BD}) = 96^\circ - 60^\circ = 36^\circ$. This matches option D. Let's check the provided solution which is A, 66. If $m(\text{arc BD}) = 66^\circ$, then $m\angle C = \frac{1}{2} (96^\circ - 66^\circ) = \frac{1}{2} (30^\circ) = 15^\circ$. This is not $30^\circ$. There might be a mistake in my understanding or the provided solution. Let's assume the angle $30^\circ$ is actually $\angle BCE$ or $\angle ACD$. If it is $\angle ACD$, then it is an inscribed angle subtending arc AD. But C is outside the circle. Let's assume the lines AC and CE are secants. The angle at C is $30^\circ$. The intercepted arcs are arc AE and arc BD. The formula is $m\angle C = \frac{1}{2} (m(\text{arc AE}) - m(\text{arc BD}))$. We are given $m(\text{arc AE}) = 96^\circ$. So $30^\circ = \frac{1}{2} (96^\circ - m(\text{arc BD}))$. $60^\circ = 96^\circ - m(\text{arc BD})$. $m(\text{arc BD}) = 96^\circ - 60^\circ = 36^\circ$. This is option D. Let's consider if the angle $30^\circ$ is $\angle ACB$. This is not an angle formed by secants. Let's consider if the angle $30^\circ$ is $\angle ECD$. This is also not directly useful. Let's assume the question is asking for the measure of arc AB or arc DE. But it clearly asks for arc BD. Let's re-examine the diagram. The lines AC and CE are secants. The angle at C is $30^\circ$. The intercepted arcs are arc AE and arc BD. The formula for the angle formed by two secants is $m\angle C = \frac{1}{2} (m(\text{far arc}) - m(\text{near arc}))$. In this case, the far arc is AE and the near arc is BD. So $30^\circ = \frac{1}{2} (m(\text{arc AE}) - m(\text{arc BD}))$. We are given $m(\text{arc AE}) = 96^\circ$. So $30^\circ = \frac{1}{2} (96^\circ - m(\text{arc BD}))$. Multiply both sides by 2: $60^\circ = 96^\circ - m(\text{arc BD})$. Rearrange to solve for $m(\text{arc BD})$: $m(\text{arc BD}) = 96^\circ - 60^\circ = 36^\circ$. This is option D.

Let's consider another possibility. What if the angle $30^\circ$ is not $\angle C$ but some other angle related to the secants? However, the diagram clearly labels the angle at C as $30^\circ$.

Let's assume there is a mistake in the problem or the options. If we assume that the angle $30^\circ$ is actually the angle subtended by arc BD at the circumference, then arc BD would be $2 \times 30^\circ = 60^\circ$. This is not an option.

Let's assume that the angle $30^\circ$ is the angle subtended by arc AE at the circumference, then arc AE would be $2 \times 30^\circ = 60^\circ$. But arc AE is given as $96^\circ$.

Let's go back to the secant formula. $30^\circ = \frac{1}{2} (96^\circ - m(\text{arc BD}))$. This leads to $m(\text{arc BD}) = 36^\circ$.

Let's consider the possibility that the angle $30^\circ$ is not $\angle C$ but some other angle. However, the diagram is quite clear.

Let's consider if the formula is applied incorrectly. The formula for the angle formed by two secants intersecting outside a circle is indeed half the difference of the intercepted arcs.

Let's consider if the arcs are mislabeled. If arc BD was the far arc and arc AE was the near arc, then $30^\circ = \frac{1}{2} (m(\text{arc BD}) - 96^\circ)$. This would mean $60^\circ = m(\text{arc BD}) - 96^\circ$, so $m(\text{arc BD}) = 156^\circ$. This is not an option.

Let's assume the provided answer A (66) is correct and work backwards. If $m(\text{arc BD}) = 66^\circ$, then $m\angle C = \frac{1}{2} (96^\circ - 66^\circ) = \frac{1}{2} (30^\circ) = 15^\circ$. This contradicts the given $30^\circ$.

There seems to be an inconsistency in the problem statement, diagram, options, or the provided solution. However, based on the ...