To answer the given problem from the image, I will break down the provided questions systematically.

The problem relates to the geometry of circles, where $E$ is the center of the circle. Key formulas and theorems such as central angles, inscribed angles, and arcs will be applied.

Step-by-Step Answers:

1. $\overarc{C}$ is inscribed in arc: $\text{Answer: } \overarc{AC}$ Explanation: The inscribed angle $\angle C$ is subtended by arc $\overarc{AC}$.

2. If $m\angle AEC = 74^\circ$, $m\overarc{AC} =$: $m\overarc{AC} = 74^\circ$ Explanation: The measure of a central angle equals the measure of the intercepted arc.

3. The angle inscribed in $\overarc{AB}$ is: $m\angle = \frac{1}{2} m\overarc{AB}$ Explanation: An inscribed angle is half the measure of the intercepted arc. If specific values for $\overarc{AB}$ were provided, you could plug them in.

4. If $m\overarc{CD} = 70^\circ$, then $m\angle A =$: $m\angle A = \frac{1}{2} m\overarc{CD} = \frac{1}{2}(70^\circ) = 35^\circ$ Explanation: $\angle A$ is an inscribed angle subtending arc $\overarc{CD}$.

5. $m\overarc{AB} +$ _______ $= m\overarc{ABD}$: $m\overarc{AB} + m\overarc{BD} = m\overarc{ABD}$ Explanation: Arc $\overarc{ABD}$ is the sum of arcs $\overarc{AB}$ and $\overarc{BD}$.

Follow-Up Questions:

1. What is the formula for finding the angle subtended by a chord at the center of a circle?
2. How does the measure of an inscribed angle compare to the intercepted arc?
3. If the total arc of a circle is $360^\circ$, how are smaller arcs calculated for specific central angles?
4. What is the relationship between opposite angles in a cyclic quadrilateral?
5. How do you calculate arc measures if only the radius and chord length are provided?

Tip:

Always remember the key relationship: central angle $=$ intercepted arc, while inscribed angle $= \frac{1}{2}$ intercepted arc. This principle simplifies many circle geometry problems!