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The given diagram shows two intersecting lines with marked angles labeled as \((2x + 52)^\circ\) and \((6x + 8)^\circ\).

Since these angles are vertically opposite, they must be equal. Thus, we can set up the equation:

\[
2x + 52 = 6x + 8
\]

Now, let's solve for \(x\):

1. Subtract \(2x\) from both sides:
\[
52 = 4x + 8
\]

2. Subtract 8 from both sides:
\[
44 = 4x
\]

3. Divide both sides by 4:
\[
x = 11
\]

Now that we know \(x = 11\), we can substitute it back into the expressions for the angles:

\[
\text{First angle} = 2x + 52 = 2(11) + 52 = 22 + 52 = 74^\circ
\]

\[
\text{Second angle} = 6x + 8 = 6(11) + 8 = 66 + 8 = 74^\circ
\]

Thus, both marked angles are \(74^\circ\).

Would you like further 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 an equation with variables on both sides?
3. What happens when two lines intersect in terms of angle relationships?
4. Can two different angles ever be vertically opposite?
5. What are supplementary and complementary angles, and how do they relate to intersecting lines?

**Tip:** Vertically opposite angles are always equal when two straight lines intersect, making them a reliable tool in solving angle-related problems.

Find the measure of the marked angles: (2x + 52)° and (6x + 8)° in an intersecting line diagram.

Learn how to solve for the measure of vertically opposite angles using algebra. This example problem shows how to set up and solve the equation 2x + 52 = 6x + 8, which demonstrates key concepts of algebra and geometry. Ideal for students in Grades 7-9.

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