The image shows two parallel lines, $m \parallel n$, with a transversal intersecting both lines. The angles formed are labeled $x^\circ$, $y^\circ$, and one angle is given as $51^\circ$. To find $x$ and $y$, we will use the properties of parallel lines and transversals.

Step-by-step solution:

1. Corresponding Angles: Since $m \parallel n$, corresponding angles are congruent. The angle labeled $51^\circ$ corresponds to the angle $y^\circ$. Therefore: $y^\circ = 51^\circ$

2. Linear Pair: The angles $x^\circ$ and $y^\circ$ are on a straight line (they form a linear pair), so they must add up to $180^\circ$: $x^\circ + y^\circ = 180^\circ$ Substituting $y^\circ = 51^\circ$: $x^\circ + 51^\circ = 180^\circ$ Solving for $x$: $x^\circ = 180^\circ - 51^\circ = 129^\circ$

Final Answer:

• $x = 129^\circ$
• $y = 51^\circ$

Would you like more details on this, or do you have any further questions?

Here are 5 related questions to further your understanding:

1. What are alternate interior angles, and how can they be used in similar problems?
2. How would the problem change if the lines were not parallel?
3. Can you prove the relationship between corresponding angles in parallel lines?
4. How can you use vertical angles to find other unknown angles in this diagram?
5. What happens to angle relationships if the transversal is perpendicular to the parallel lines?

Tip: Always remember that in parallel line setups, corresponding angles are equal, and linear pairs sum to $180^\circ$.