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.