https://r2.erweima.ai/mathbot/image/1726601144394sn5zemdl.jpg

In the image, we have a circle with a central point \( O \) and two angles inside the circle. There are two given angles marked:

- \( \alpha + 10^\circ \)
- \( \alpha + 60^\circ \)

Both angles seem to have their vertices at point \( O \), suggesting that they are central angles of the circle.

To solve this, we can use the geometric property that the sum of all central angles around a point \( O \) in a circle is \( 360^\circ \). Therefore, we have the following equation:

\[
(\alpha + 10^\circ) + (\alpha + 60^\circ) = 360^\circ
\]

Now, let's solve this equation:

\[
2\alpha + 70^\circ = 360^\circ
\]

\[
2\alpha = 360^\circ - 70^\circ
\]

\[
2\alpha = 290^\circ
\]

\[
\alpha = \frac{290^\circ}{2} = 145^\circ
\]

Thus, \( \alpha = 145^\circ \).

Let me know if you need more details or further clarifications!

### Related Questions:
1. What are central angles in a circle, and how do they relate to the circumference?
2. Can central angles be more than 180°?
3. What is the difference between central angles and inscribed angles?
4. How does the measure of an arc relate to its central angle?
5. What happens to the angles when a circle is divided into three or more sections?

### Tip:
Always remember that the sum of the angles around a point (like the center of a circle) is always \(360^\circ\), which is useful for solving such problems.

Given a circle with central point O and two angles inside the circle, labeled α + 10° and α + 60°, solve for the value of α.

Learn how to solve for α when given two central angles α + 10° and α + 60° in a circle. The problem covers key geometric concepts such as central angles and the sum of angles around a point. Suitable for students in Grades 9-11.

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