The image shows two parallel lines $m \parallel n$ with a transversal intersecting them. You are asked to find the values of $x$ and $y$.

Here are the key pieces of information:

• One angle on line $m$ is labeled $(9x - 12)^\circ$.
• The angle adjacent to it on the same line is labeled $y^\circ$.
• The angle on line $n$, corresponding to the transversal angle, is $(5x - 4)^\circ$.

Solution Steps:

Since the lines $m \parallel n$ are parallel, we can use properties of angles formed by a transversal:

1. Corresponding Angles: Angles on the same side of the transversal and in matching positions (like $(9x - 12)^\circ$ and $(5x - 4)^\circ$) are equal.

Thus, we can set the angles equal to each other:

$9x - 12 = 5x - 4$

Solve for $x$:

$9x - 5x = -4 + 12$ $4x = 8$ $x = 2$

2. Finding $y$: Now that we know $x = 2$, substitute it into $(9x - 12)^\circ$ to find that angle:

$9(2) - 12 = 18 - 12 = 6^\circ$

Thus, the angle on line $m$ is $6^\circ$, which means $y = 6^\circ$, since those two angles are adjacent and supplementary (adding up to 180°).

Final Answer:

• $x = 2$
• $y = 6^\circ$

Would you like further details on how we used corresponding angles or any other part of the solution?

Here are five related questions you might explore:

1. What are alternate interior angles and how do they relate to parallel lines?
2. How would the solution change if the lines were not parallel?
3. What are supplementary angles and how do they apply here?
4. Can you find the sum of interior angles for different types of polygons?
5. What role do transversal lines play in geometry with angles?

Tip: When solving problems involving parallel lines, always look for corresponding or alternate interior angles to simplify your calculations.