The problem provided involves a circle with center $O$, and it states:

1. $\angle AOE = 117^\circ$.
2. $\angle BOD = 37^\circ$.
3. We are tasked with finding $\angle BCD$.

Solution:

Since $\angle BOD$ is the central angle, it subtends $\angle BCD$ at the circumference.

1. Property of Angles in a Circle: The angle subtended by an arc at the center of a circle is twice the angle subtended at the circumference. Therefore: $\angle BCD = \frac{1}{2} \times \angle BOD$

2. Substitute the given value of $\angle BOD = 37^\circ$: $\angle BCD = \frac{1}{2} \times 37^\circ = 18.5^\circ$

Thus, the measure of $\angle BCD$ is $18.5^\circ$.

Would you like a detailed explanation on how angles in circles work or help with other related problems?

Here are 5 related questions you may find useful:

1. What is the relationship between central and inscribed angles in a circle?
2. How do you calculate the arc length of a circle given the central angle?
3. Can a central angle ever be less than its inscribed angle?
4. How do angles subtended by the same arc relate to each other?
5. What is the sum of opposite angles in a cyclic quadrilateral?

Tip: Always remember, the central angle is twice any angle it subtends on the circumference when they open to the same arc.