In this image, you are given two parallel lines $l \parallel m$ and angles formed by a transversal. The goal is to find the values of $x$ and $y$.

We have the following angles:

• $(3x - 16)^\circ$ and $(6x + 7)^\circ$ are alternate interior angles.
• $(11y - 32)^\circ$ and $(6x + 7)^\circ$ are corresponding angles.

Since $l \parallel m$, alternate interior angles are equal, and corresponding angles are also equal.

Step 1: Set up the equations for alternate interior angles

Since $(3x - 16)^\circ$ and $(6x + 7)^\circ$ are alternate interior angles: $3x - 16 = 6x + 7$ Solve for $x$: $3x - 6x = 7 + 16$ $-3x = 23$ $x = -\frac{23}{3}$

Step 2: Set up the equation for corresponding angles

Since $(11y - 32)^\circ$ and $(6x + 7)^\circ$ are corresponding angles, they are equal: $11y - 32 = 6x + 7$ Substitute $x = -\frac{23}{3}$ into this equation: $11y - 32 = 6\left(-\frac{23}{3}\right) + 7$ $11y - 32 = -46 + 7$ $11y - 32 = -39$ Solve for $y$: $11y = -39 + 32$ $11y = -7$ $y = -\frac{7}{11}$

Final Answer:

• $x = -\frac{23}{3}$
• $y = -\frac{7}{11}$

Let me know if you want more details or have any questions.

Here are some related questions you might explore:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. How do corresponding angles behave when two lines are parallel?
3. What is the significance of a transversal in geometry?
4. How can equations involving angles help determine unknown variables?
5. What other types of angle relationships can be formed by parallel lines and transversals?

Tip: Always check if angles are alternate, corresponding, or supplementary when dealing with parallel lines and a transversal.