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.