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The problem you've provided shows two angles that are vertically opposite, which means they are equal. The expressions for the angles are \( (x + 7)^\circ \) and \( (5x - 49)^\circ \).

Since vertically opposite angles are equal, we can set the expressions equal to each other:

\[
x + 7 = 5x - 49
\]

### Solving for \( x \):

1. Subtract \( x \) from both sides:

\[
7 = 4x - 49
\]

2. Add 49 to both sides:

\[
56 = 4x
\]

3. Divide by 4:

\[
x = 14
\]

### Finding the angle measures:

- For the first angle: \( x + 7 = 14 + 7 = 21^\circ \)
- For the second angle: \( 5x - 49 = 5(14) - 49 = 70 - 49 = 21^\circ \)

Thus, both angles are \( 21^\circ \).

Would you like more details or have any questions? Here are some related problems for practice:

1. What are vertically opposite angles, and why are they always equal?
2. How would the problem change if the angles were supplementary instead of equal?
3. If the angles were on the same side of a transversal line, what relationship would they have?
4. Can you create a similar problem but use complementary angles?
5. How would the solution differ if one of the expressions included a squared term?

**Tip:** When solving linear equations, isolating the variable by systematically using inverse operations helps simplify the process.

Find the value of the variable and the measure of each labeled angle. Given the angles (x + 7)° and (5x - 49)° are vertically opposite angles.

Learn how to solve for x when given two vertically opposite angles, (x + 7)° and (5x - 49)°. This problem demonstrates key algebraic and geometric concepts such as angle relationships. Suitable for students in Grades 7-9.

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