To solve for $x$ and $y$ in the given diagram, we’ll analyze the angles and apply relevant geometric rules. Here’s the step-by-step reasoning:

Step 1: Identify Parallel Lines and Corresponding Angles

1. In the diagram, we see parallel lines indicated by the arrows: $AI \parallel JH$ and $CF \parallel JD$.
2. Because these lines are parallel, we can use the concept of alternate interior angles and corresponding angles.

Step 2: Analyze Angle Relationships

1. Angle $x$:

   • Since $AI \parallel JH$ and $GJ$ acts as a transversal line, we observe that the angle $\angle AIG$ (which is $70^\circ$) and $\angle x$ are alternate interior angles.
   • Therefore, by the alternate interior angles theorem, we can conclude: $x = 70^\circ$

2. Angle $y$:

   • Now, focusing on the angles at point $J$ where $JD \parallel CF$ and $CJ$ is a transversal.
   • Since $\angle JBD = 50^\circ$, it forms an alternate interior angle with $\angle y$.
   • Therefore, by the alternate interior angles theorem again, we conclude: $y = 50^\circ$

Solution Summary

• $x = 70^\circ$ (Alternate Interior Angles)
• $y = 50^\circ$ (Alternate Interior Angles)

Would you like more detailed explanations or have any further questions?

Here are five related questions to expand on this topic:

1. How do we identify corresponding angles in parallel line diagrams?
2. What is the difference between alternate interior and corresponding angles?
3. How would you solve for unknown angles if there were no given measurements?
4. How do transversal lines affect angle calculations?
5. Why is it essential to state reasons for angle equality in geometric proofs?

Tip: Always look for parallel line markings in diagrams, as they often provide clues for using angle properties like alternate interior or corresponding angles.