The diagram shows two parallel lines, $m \parallel n$, intersected by a transversal. Given the angles formed, we can use properties of parallel lines and transversals to find the values of $x$ and $y$.

Step 1: Identify relationships between angles

Since the lines $m$ and $n$ are parallel and the transversal creates alternate interior angles, we have the following relationships:

1. The angle $(6x + 19)^\circ$ is alternate interior to the angle $(7x + 15)^\circ$, so: $6x + 19 = 7x + 15$

2. The angle $y^\circ$ is corresponding to the angle $(6x + 19)^\circ$, so: $y = 6x + 19$

Step 2: Solve for $x$

Using the equation from the alternate interior angles: $6x + 19 = 7x + 15$ Subtract $6x$ from both sides: $19 = x + 15$ Subtract 15 from both sides: $x = 4$

Step 3: Solve for $y$

Now substitute $x = 4$ into the equation for $y$: $y = 6(4) + 19 = 24 + 19 = 43$

Final Answer:

• $x = 4$
• $y = 43^\circ$

Would you like further explanation or details on this process? Here are some related questions:

1. What are alternate interior angles, and why do they have equal measures?
2. Can corresponding angles be used to solve for both unknowns in this problem?
3. How can we verify that the solution satisfies both relationships in the diagram?
4. What other angle properties apply when dealing with parallel lines and a transversal?
5. How can we use supplementary angles to check the consistency of the results?

Tip: Always verify the relationships between angles when working with parallel lines and a transversal, as multiple angle properties (alternate, corresponding, supplementary) can simplify the solution.