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In the diagram below, line d 
∥ line e and line a 
∥ line b.  Determine the values of x and y.

Let's analyze the given diagram and use geometric relationships to solve for \( x \) and \( y \).

### Step 1: Identify relationships between angles
We are told that:
- Line \( d \parallel e \)
- Line \( a \parallel b \)

Using properties of parallel lines and transversal lines, we can apply the **Alternate Interior Angle Theorem** and the **Corresponding Angle Theorem**.

### Step 2: Analyzing the angles in the diagram

1. The angle marked \( x^\circ \) and the angle marked \( 5x^\circ \) are **corresponding angles** because line \( a \parallel b \) and the transversal is the slanted line through \( c \). Hence, these angles are equal:
   \[
   x = 5x
   \]

2. The angle marked \( 2y^\circ \) and the angle marked \( (2x + 5)^\circ \) are also **corresponding angles** because line \( b \parallel a \), and the transversal is the slanted line through \( d \). Hence, these angles are equal:
   \[
   2y = 2x + 5
   \]

### Step 3: Solve the system of equations

We now have two equations:
1. \( x = 5x \)
2. \( 2y = 2x + 5 \)

Let me solve this system to find \( x \) and \( y \).The solution to the system of equations is:

- \( x = 0 \)
- \( y = \frac{5}{2} \) or \( y = 2.5 \)

Thus, the values of \( x \) and \( y \) are \( 0 \) and \( 2.5 \), respectively.

Would you like more details on any specific step? Here are some related questions to explore:

1. How do corresponding angles work in parallel lines?
2. What is the Alternate Interior Angle Theorem?
3. How do transversal lines affect angle relationships?
4. Could we solve this problem differently, using triangle properties?
5. How do the sum of angles in a triangle influence solutions in geometry?

**Tip**: In geometry, recognizing parallel lines and transversal angles is key to solving complex diagrams. Always look for corresponding, alternate, and co-interior angles!

In the diagram, line d ∥ line e and line a ∥ line b. Determine the values of x and y.

Solve for x and y in a geometry problem involving parallel lines and angle relationships. Explore how the Alternate Interior Angle Theorem and Corresponding Angle Theorem apply. Suitable for students in Grades 7-9.

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