https://file.aiquickdraw.com/mathbot/file/1726595163357p76paand.jpg

From the image, we have two intersecting lines that form a system of angles. The labeled angles are \( 3x^\circ \) and \( (76 - x)^\circ \), and since these angles are vertically opposite, they are equal. So, we can set up the following equation:

\[
3x = 76 - x
\]

Let's solve for \( x \):

\[
3x + x = 76
\]
\[
4x = 76
\]
\[
x = \frac{76}{4} = 19
\]

Now, substitute \( x = 19 \) back into the expression for the angle \( 3x \):

\[
3x = 3 \times 19 = 57^\circ
\]

Thus, the measure of the angle \( 3x^\circ \) is \( 57^\circ \).

Would you like more details or have any other questions?

Here are 5 related questions:
1. What are vertically opposite angles, and why are they equal?
2. How do you solve equations involving multiple variables?
3. What is the measure of the angle \( (76 - x)^\circ \) when \( x = 19 \)?
4. How can we verify the consistency of these results?
5. What are complementary and supplementary angles?

**Tip:** Always check if two angles are vertically opposite when two lines intersect; it helps simplify solving for unknowns.

Find the measures of the labeled angles. The angles are 3x degrees and (76 - x) degrees, which are vertically opposite.

Learn how to solve for labeled angles using the concept of vertically opposite angles. The problem involves solving the equation 3x = 76 - x, leading to x = 19 and an angle of 57 degrees. Suitable for Grades 6-8.

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