https://file.aiquickdraw.com/mathbot/file/1725915712094xb22ivzo.jpg

In this problem, we are working with a circle where \( O \) is the center. There are two angles given: \( 50^\circ \) and \( 60^\circ \), along with two unknown angles \( x \) and \( y \). We need to calculate the value of \( x + y \).

### Observations:

- \( x \) is an angle at the center of the circle formed by two radii.
- \( y \) is an angle at the circumference subtended by the same arc that forms angle \( x \).

#### Key Properties:

1. **Angle at the center** \( x \) is twice the angle at the circumference subtended by the same arc. This is known as the *angle at the center theorem*. So, \( x = 2y \).
   
2. The total sum of angles in the quadrilateral formed inside the circle is \( 360^\circ \). Since we have angles \( 50^\circ \), \( 60^\circ \), and the center angle \( x \), the remaining angle at the circumference is \( y \).

### Steps to solve:
1. Apply the angle at the center theorem: \( x = 2y \).
2. Sum of angles around the center of the circle is \( 360^\circ \). Use this fact to calculate \( x + y \).

Now, let's proceed with the calculations.

- Sum of the angles in the quadrilateral: \( 50^\circ + 60^\circ + x + y = 180^\circ \).

Let's calculate this step by step.The solution for the angles is:

- \( x = \frac{140}{3} \) degrees, or approximately \( 46.67^\circ \)
- \( y = \frac{70}{3} \) degrees, or approximately \( 23.33^\circ \)

Thus, the value of \( x + y \) is:

\[
x + y = \frac{140}{3} + \frac{70}{3} = 70^\circ
\]

Would you like more details on the solution process or have any questions?

Here are 5 related questions:
1. How does the angle at the center theorem apply to other circle properties?
2. What is the general method for solving quadrilateral angle problems in a circle?
3. How can the sum of angles be used in complex circle theorems?
4. Can the angle at the center theorem be extended to non-circular shapes?
5. What are other key theorems involving angles in circles?

**Tip:** Remember that angles subtended by the same arc at the circumference are equal, a useful property when dealing with circle theorems.

In the following figure, O is the center of the circle. Given the angles 50° and 60°, calculate the value of x + y.

This problem involves applying the angle at the center theorem in a circle, where two angles 50° and 60° are given. Learn how to calculate the value of x + y using the key properties of circle geometry. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation