In the image provided, you are given two parallel lines, $m \parallel n$, and asked to find the values of $x$ and $y$.

The given angles are:

• $(x + 19)^\circ$ (at the top right)
• $(3y + 8)^\circ$ (at the bottom left)
• $(2x + 1)^\circ$ (adjacent to the previous angle)

Key Concept:

Since the lines $m$ and $n$ are parallel, several angle relationships apply. Specifically, alternate interior angles are congruent, and corresponding angles are congruent.

Step 1: Use the corresponding angles relationship.

The angle $(x + 19)^\circ$ at the top is corresponding to $(2x + 1)^\circ$ at the bottom. Since they are corresponding angles, they must be equal: $x + 19 = 2x + 1$ Solve for $x$: $x + 19 = 2x + 1$ Subtract $x$ from both sides: $19 = x + 1$ Subtract 1 from both sides: $x = 18$

Step 2: Use the alternate interior angles relationship.

The angle $(3y + 8)^\circ$ is an alternate interior angle to $(x + 19)^\circ$. Since alternate interior angles are equal: $3y + 8 = x + 19$ Substitute $x = 18$ from the previous step: $3y + 8 = 18 + 19$ $3y + 8 = 37$ Solve for $y$: $3y = 37 - 8$ $3y = 29$ $y = \frac{29}{3} = 9.67$

Final Answer:

• $x = 18$
• $y \approx 9.67$

Let me know if you'd like further clarification!

Related Questions:

1. How do we derive the alternate interior angles theorem?
2. What is the sum of interior angles for a set of parallel lines and a transversal?
3. Can the corresponding angles theorem be applied to non-parallel lines?
4. How would the problem change if the lines were not parallel?
5. What other geometric properties can be derived from parallel lines and transversals?

Tip:

When working with parallel lines and transversals, always remember to check for angle relationships like alternate interior, corresponding, and vertically opposite angles to simplify the problem.