In this diagram, two lines $l$ and $m$ are intersected by a transversal. The angles formed on opposite sides of the transversal are given as:

• $(9x + 25)^\circ$ on line $l$,
• $(13x - 19)^\circ$ on line $m$.

These angles appear to be corresponding angles since they occupy the same relative position at the intersection points. For two corresponding angles to be equal, the lines $l$ and $m$ must be parallel.

Thus, we can set the angles equal to each other and solve for $x$:

$9x + 25 = 13x - 19$

Let's solve this equation.

$9x + 25 = 13x - 19$ Subtract $9x$ from both sides: $25 = 4x - 19$ Add 19 to both sides: $44 = 4x$ Divide by 4: $x = 11$

Now that we have $x = 11$, we can substitute this value back into the expressions for the angles to verify:

• $(9x + 25) = 9(11) + 25 = 99 + 25 = 124^\circ$,
• $(13x - 19) = 13(11) - 19 = 143 - 19 = 124^\circ$.

Since both angles are indeed equal, our solution is correct. Therefore, $x = 11$.

Would you like any further explanation or clarification?

Here are 5 related questions:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. How can you identify corresponding angles in a transversal setup?
3. What is the significance of the transversal line in geometry?
4. How do we know the lines $l$ and $m$ are parallel in this situation?
5. Could these angles be supplementary instead, and how would that change the problem?

Tip: When solving for variables in geometry, always check whether the angle relationships are equal (like corresponding angles) or supplementary (like consecutive interior angles).