https://file.aiquickdraw.com/mathbot/file/1727688412443spq9ixz1.jpg

We are given a circle with the following conditions:

- \( \overline{BE} \) is the diameter.
- \( m \widehat{CD} = 40^\circ \)
- \( m \widehat{DE} = 100^\circ \)
- \( m \widehat{EF} = 120^\circ \)

Let's analyze the angles one by one, referring to the diagram:

1. **\( m \angle 1 ( \angle FAE ) \)**:
   - \( \angle FAE \) is an inscribed angle subtended by arc \( FE \). The arc \( FE \) has a measure of 120° (given).
   - The measure of an inscribed angle is half the measure of its intercepted arc:
     \[
     m \angle 1 = \frac{1}{2} \times 120^\circ = 60^\circ
     \]

2. **\( m \angle 2 \)**:
   - \( \angle 2 \) is subtended by the diameter \( BE \). By the inscribed angle theorem, any angle subtended by a diameter is a right angle (90°):
     \[
     m \angle 2 = 90^\circ
     \]

3. **\( m \angle 3 ( \angle BAE ) \)**:
   - \( \angle BAE \) is subtended by arc \( BE \), which is a semicircle (180° because \( BE \) is the diameter).
   - The measure of an inscribed angle is half the measure of its intercepted arc:
     \[
     m \angle 3 = \frac{1}{2} \times 180^\circ = 90^\circ
     \]

4. **\( m \angle 4 \)**:
   - \( \angle 4 \) is an exterior angle at point \( O \), subtended by arc \( EF \). Since we know \( \widehat{EF} = 120^\circ \), this will be an angle related to the arc measure.
   - This angle can be solved geometrically based on the exterior angle theorem, and it appears to be \( 90^\circ \) because it complements angle \( \angle 2 \).

5. **\( m \angle 5 \)**:
   - \( \angle 5 \) is subtended by arc \( CD \), whose measure is \( 40^\circ \).
   - The inscribed angle is half of the intercepted arc:
     \[
     m \angle 5 = \frac{1}{2} \times 40^\circ = 20^\circ
     \]

6. **\( m \angle 6 \)**:
   - \( \angle 6 \) is subtended by arc \( DE \), which has a measure of \( 100^\circ \).
   - The inscribed angle is half of the intercepted arc:
     \[
     m \angle 6 = \frac{1}{2} \times 100^\circ = 50^\circ
     \]

### Final Answers:

25. \( m \angle 1 = 60^\circ \)  
26. \( m \angle 2 = 90^\circ \)  
27. \( m \angle 3 = 90^\circ \)  
28. \( m \angle 4 = 90^\circ \)  
29. \( m \angle 5 = 20^\circ \)  
30. \( m \angle 6 = 50^\circ \)

Would you like further details on any of these steps?

---

Here are 5 related questions to expand your understanding:

1. How does the inscribed angle theorem help in solving problems in circles?
2. Why is any angle subtended by a diameter always \( 90^\circ \)?
3. What are exterior and interior angles in a circle, and how do they relate to intercepted arcs?
4. How can arc measures help in finding angles within a cyclic quadrilateral?
5. Can the angles in a semicircle always be right angles, regardless of other angles in the circle?

**Tip:** When solving problems involving circles, always look for relationships between angles and arcs using the inscribed angle theorem and other circle theorems!

Find the measure of each of the numbered angles in the circle with given arcs and a diameter.

Learn how to find the measures of various angles in a circle using the inscribed angle theorem, arc measures, and the angle subtended by the diameter. This problem includes detailed solutions involving multiple circle theorems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation