The image you have provided depicts two parallel lines with a transversal, and there are algebraic expressions for the angles formed. Based on the geometry of the diagram, these angles are either corresponding angles or alternate interior angles, which means they should be congruent.

The expressions for the angles are:

• One angle is $13x + 13$
• The other angle is $14x + 5$

Since these are corresponding or alternate interior angles, we can set the two expressions equal to each other: $13x + 13 = 14x + 5$

Let's solve for $x$:

Step 1: Subtract $13x$ from both sides:

$13x + 13 - 13x = 14x + 5 - 13x$ $13 = x + 5$

Step 2: Subtract 5 from both sides:

$13 - 5 = x$ $x = 8$

Thus, $x = 8$.

Conclusion:

The value of $x$ is 8.

Would you like more details on the geometry behind this, or do you have any other questions?

Follow-up Questions:

1. What are corresponding angles, and why are they equal in parallel lines?
2. How do alternate interior angles relate to parallel lines and a transversal?
3. How would the solution change if the lines were not parallel?
4. What happens when we solve for $x$ if the angle expressions were given as unequal (e.g., different angles)?
5. How can we verify the solution by substituting $x = 8$ into the expressions?

Tip:

When working with angles formed by parallel lines and a transversal, always check for angle relationships like corresponding or alternate interior angles—they are often equal, simplifying calculations.