The diagram shows two parallel lines, $l$ and $m$, cut by a transversal, forming two angles labeled $(8x - 77)^\circ$ and $(3x + 38)^\circ$. Here's how we can address the questions:

1. What angles are given?

The angles $(8x - 77)^\circ$ and $(3x + 38)^\circ$ are on opposite sides of the transversal and inside the parallel lines $l$ and $m$. This makes them alternate interior angles.

2. Solve for $x$.

Since the angles are alternate interior angles, they are equal when the lines are parallel. So, we set the expressions equal to each other:

$8x - 77 = 3x + 38$

Now, solve for $x$:

1. Subtract $3x$ from both sides:

   $8x - 3x - 77 = 38$

   $5x - 77 = 38$

2. Add 77 to both sides:

   $5x = 115$

3. Divide by 5:

   $x = 23$

Solution

• The given angles are alternate interior angles.
• The value of $x$ is 23.

Would you like further details or have any questions?

Related Questions

1. What are alternate exterior angles, and how do they differ from alternate interior angles?
2. How do you identify corresponding angles in a transversal diagram?
3. What other angle relationships exist when two parallel lines are cut by a transversal?
4. Can consecutive interior angles be supplementary? Explain why.
5. How would the solution change if the lines were not parallel?

Tip

When working with parallel lines and a transversal, remember that alternate interior angles are always equal, which is a helpful property for solving equations.