https://file.aiquickdraw.com/mathbot/file/1726044108081lh5pkt0a.jpg

In this image, we have a geometric figure with two parallel lines, \( AB \parallel CD \), and a transversal line intersecting these parallel lines at point \( E \). The angles given are \( \angle AEC = 45^\circ \) and \( \angle BED = 60^\circ \). We are tasked with finding the angles \( x \) and \( y \).

### Step-by-Step Solution:

1. **Using the fact that AB is parallel to CD:**
   Since \( AB \parallel CD \) and \( AE \) and \( ED \) are transversals, the following properties apply:

2. **Angles on a straight line:**
   - At point \( E \), the angles on the straight line \( AB \) sum up to 180°. Therefore:
   \[
   \angle AEB = \angle AEC + \angle BED = 45^\circ + 60^\circ = 105^\circ
   \]
   So, the remaining angle is:
   \[
   \angle AEB = 180^\circ - 105^\circ = 75^\circ
   \]

3. **Using the corresponding angles property:**
   Since \( AB \parallel CD \), the angle \( x \) is corresponding to \( \angle AEB \). Therefore:
   \[
   x = \angle AEB = 75^\circ
   \]

4. **Using the sum of angles in a triangle:**
   The sum of angles in triangle \( CED \) is 180°. Thus:
   \[
   \angle CED + x + y = 180^\circ
   \]
   Substituting the known values:
   \[
   45^\circ + 75^\circ + y = 180^\circ
   \]
   Solving for \( y \):
   \[
   y = 180^\circ - 120^\circ = 60^\circ
   \]

### Final Answers:
- \( x = 75^\circ \)
- \( y = 60^\circ \)

Would you like more details on any part of the solution or clarification?

Here are 5 related questions to expand this:
1. What are alternate interior angles, and how do they relate to parallel lines?
2. How do corresponding angles work when a transversal intersects parallel lines?
3. Can you solve for unknown angles in a different triangle if given two angles?
4. What are the properties of supplementary angles?
5. How does the sum of angles in polygons change as the number of sides increases?

**Tip:** When working with parallel lines and transversals, remember that corresponding, alternate interior, and consecutive interior angles are essential tools for solving problems involving unknown angles.

In the given figure, AB || CD. Find x and y.

Learn how to solve for x and y in a geometry problem involving parallel lines AB || CD, angles of 45° and 60°, and the sum of angles in a triangle. This problem helps students in Grades 8-10 understand the concept of corresponding angles and how to use them.

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