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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 \( \overarc{BD} \).

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### 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!

Explore a step-by-step solution for evaluating central and inscribed angles in circle geometry. This detailed explanation covers key concepts like arc measures, the Central Angle Theorem, and the Inscribed Angle Theorem. Suitable for Grades 8-10.

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