https://file.aiquickdraw.com/mathbot/file/1726757920060x62sj94d.jpg

In the given image, the problem asks to find the value of \(x\), and a diagram is provided. Here's how we can solve this:

1. The diagram shows two intersecting lines forming vertical angles. One of the angles is labeled \(x\), and another angle is labeled \(\frac{x}{2}\) and 50°.
   
2. The key geometric property to use here is that **the sum of vertically opposite angles is equal**, and **the sum of angles on a straight line is 180°**.

3. According to the diagram:
   - The angle \(x\) and the angle \(\frac{x}{2} + 50^\circ\) form a linear pair (since they are on a straight line), and their sum must be 180°.

### Equation Setup:
\[
x + \left(\frac{x}{2} + 50\right) = 180
\]

### Solving for \(x\):
\[
x + \frac{x}{2} + 50 = 180
\]
\[
x + \frac{x}{2} = 180 - 50
\]
\[
x + \frac{x}{2} = 130
\]

Now, multiply the entire equation by 2 to eliminate the fraction:
\[
2x + x = 260
\]
\[
3x = 260
\]
\[
x = \frac{260}{3} = 86.67^\circ
\]

### Conclusion:
The measure of \(x\) is approximately \(86.67^\circ\).

Would you like further details or have any questions?

### Related Questions:
1. How do you prove the vertical angle theorem?
2. What is the significance of supplementary angles?
3. Can two adjacent angles both be acute on a straight line?
4. What is the difference between linear pairs and vertically opposite angles?
5. How would the calculation change if the angle was given as \(\frac{2x}{3} + 50^\circ\)?

### Tip:
Always check the relationships between angles in geometry problems, especially when dealing with intersecting lines, to set up correct equations.

Find the measure of x given the intersecting lines with angles x and (x/2 + 50°).

Learn how to find the measure of x in intersecting lines using linear pairs and vertically opposite angles. This problem covers key geometric concepts, such as the Linear Pair Theorem and the Vertical Angles Theorem, suitable for Grades 7-9.

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