I see that you've uploaded an image related to measuring arcs or central angles in a circle. Let me analyze the image and provide the necessary solutions. I'll extract the given angles and calculate the required measures.

Give me a moment to process this.From the image, we see several circle diagrams with given angles. The problem asks us to find the measure of the arc or central angle, assuming that the lines appearing to be diameters are actual diameters.

Step-by-Step Solution:

1. Understanding Diameters:

   • A diameter divides the circle into two equal semicircles of 180°.
   • If an angle is given as part of the circle, we can use this information to determine the missing angles.

2. Solving Each Case:

   • Figure 1: The given angle is 140°. Since a diameter forms a semicircle, the remaining arc would be 180° - 140° = 40°.
   • Figure 2: The given central angle is 110°. The remaining arc in the semicircle is 180° - 110° = 70°.
   • Figure 3: The given angles are 75° and 90°. The sum is 75° + 90° = 165°, so the remaining arc is 180° - 165° = 15°.
   • Figure 4: Given angles are 65° and 85°. Their sum is 65° + 85° = 150°, so the remaining arc is 180° - 150° = 30°.

These calculations provide the measures of the arcs or central angles needed.

Would you like further explanations or details? Here are some related questions:

1. How do we determine the arc length given a radius?
2. What is the relationship between a central angle and its intercepted arc?
3. How do you calculate angles in a cyclic quadrilateral?
4. Can you derive the total interior angles of a polygon inscribed in a circle?
5. How do supplementary angles help in solving circle theorems?

Math Tip:

In a circle, the central angle is always equal to the measure of the intercepted arc. If two central angles share a common arc, their sum is always 360°.