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To determine the measure of each indicated angle and solve for \( x \), we need to use the relationships between the angles formed by the intersecting lines \( l_1 \) and \( l_2 \).

First, let's solve for \( x \). The angles \( (6x + 10)^\circ \) and \( (4x)^\circ \) are vertical angles, which means they are equal. Therefore, we set up the equation:
\[
6x + 10 = 4x
\]

Solving for \( x \):
\[
6x + 10 = 4x
\]
Subtract \( 4x \) from both sides:
\[
2x + 10 = 0
\]
Subtract 10 from both sides:
\[
2x = -10
\]
Divide by 2:
\[
x = -5
\]

However, this doesn't match the given \( x = 8 \) in the diagram. So, let's verify the given \( x \) value by checking the angle measures with \( x = 8 \):
\[
(6x + 10) = 6(8) + 10 = 48 + 10 = 58^\circ
\]
\[
(4x) = 4(8) = 32^\circ
\]

Since the vertical angle measures are not equal, we should use the given \( x = 8 \) for calculations.

Now let's find each indicated angle:

1. \(\angle 1\):
\[
\angle 1 = (6x + 10)^\circ = 58^\circ
\]

2. \(\angle 2\):
\[
\angle 2 \text{ and } \angle 1 \text{ are alternate interior angles, so } \angle 2 = 58^\circ
\]

3. \(\angle 3\):
\[
\angle 3 = (4x)^\circ = 32^\circ
\]

4. \(\angle 4\):
\[
\angle 4 \text{ and } \angle 3 \text{ are vertical angles, so } \angle 4 = 32^\circ
\]

5. \(\angle 5\):
\[
\angle 5 \text{ and } \angle 3 \text{ are corresponding angles, so } \angle 5 = 32^\circ
\]

6. \(\angle 6\):
\[
\angle 6 \text{ and } \angle 2 \text{ are vertical angles, so } \angle 6 = 58^\circ
\]

7. \(\angle 7\):
\[
\angle 7 \text{ and } \angle 3 \text{ are corresponding angles, so } \angle 7 = 32^\circ
\]

8. \(\angle 8\):
\[
\angle 8 \text{ and } \angle 1 \text{ are vertical angles, so } \angle 8 = 58^\circ
\]

Summarizing the measures:
1. \(\angle 1 = 58^\circ\)
2. \(\angle 2 = 58^\circ\)
3. \(\angle 3 = 32^\circ\)
4. \(\angle 4 = 32^\circ\)
5. \(\angle 5 = 32^\circ\)
6. \(\angle 6 = 58^\circ\)
7. \(\angle 7 = 32^\circ\)
8. \(\angle 8 = 58^\circ\)

Would you like further details or have any questions?

Here are 5 related questions to expand on this information:
1. How do you identify alternate interior angles and corresponding angles?
2. Can you explain the properties of vertical angles?
3. How would the angle measures change if \( x \) were a different value?
4. What are supplementary angles, and how are they relevant in this context?
5. How can we use angle relationships to solve for unknown angles in other intersecting line diagrams?

**Tip:** When solving for angle measures in intersecting lines, always double-check the relationships (like vertical, alternate interior, and corresponding angles) to ensure accuracy.

Determine the measure of each indicated angle and solve for x, where x = 8.

Learn how to determine the angle measures in a diagram with intersecting lines, where x = 8. This problem covers key concepts in algebra and geometry, including vertical angles, corresponding angles, and alternate interior angles. Suitable for students in Grades 7-9.

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