To solve this problem, we need to analyze the circle $O$ and the information given. Let's go through each part of the question step-by-step.

Given Information:

• The circle $O$ has certain angle measures:
  • $m \angle 1 < m \angle 2$.
  • $m \angle 1 = m \angle 2$.
• We have to find the measures of specific angles or arc measures as listed.

Analyzing the Figure:

1. Arc and Angle Relationships:

   • There’s a central angle $\angle AOB$, which appears to have an associated measure of $70^\circ$.
   • There’s an inscribed angle $\angle ACB$ with a measure of $30^\circ$.
   • There’s also a right angle, $m \angle EDF = 90^\circ$, indicating that $\overline{EF}$ is a diameter of the circle.

2. Using Properties of Angles in a Circle:

   • The measure of an inscribed angle is half the measure of the arc it intercepts.
   • A central angle is equal to the measure of the arc it intercepts.

Finding Each Measure:

1. $m \angle \text{DEF}$:

• Since $\overline{EF}$ is the diameter, $\angle \text{DEF}$ is a right angle.
• Therefore, $m \angle \text{DEF} = 90^\circ$.

2. $m \angle 2$:

• The exact measure of $m \angle 2$ depends on further details not visible in the image or based on information regarding its relationship with angle $1$.
• Since $m \angle 1 = m \angle 2$, we might infer that they are equal, but without additional information, this part might remain undefined.

3. $m \angle 1$:

• This angle also depends on further data that might not be completely visible.

4. $m \angle \text{BAE}$:

• This angle might relate to the angles shown in the diagram, but since $m \angle ACB = 30^\circ$, it could help in determining the measure if more details were visible.

5. Sum of angles:

• Further analysis may be needed if additional parts of the diagram are clarified.

Would you like more details on how to calculate specific measures based on assumptions, or if any parts are unclear?

Follow-up Questions:

1. How do central and inscribed angles in a circle relate to the arcs they intercept?
2. What is the measure of an angle inscribed in a semicircle?
3. How do you find the measure of an angle formed by two intersecting chords?
4. Can you explain the relationship between angle $\angle AOB$ and the intercepted arc $AB$?
5. Why is $m \angle DEF$ always $90^\circ$ if $\overline{EF}$ is the diameter?

Tip:

In circle problems, always look for relationships between central angles, inscribed angles, and the arcs they intercept, as they often hold the key to solving these questions.