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We have a diagram that involves two parallel lines and transversals, which creates pairs of corresponding angles.

### Step-by-step analysis:

1. The angles \( 128^\circ \) and \( 3x + y \) are consecutive interior angles formed by the transversal. According to the consecutive interior angles theorem, the sum of these angles is \( 180^\circ \). Therefore, we have:
   \[
   128^\circ + (3x + y)^\circ = 180^\circ
   \]
   Simplifying:
   \[
   3x + y = 180^\circ - 128^\circ = 52^\circ \quad \text{(Equation 1)}
   \]

2. Similarly, the angles \( 148^\circ \) and \( 3x - y \) are also consecutive interior angles. So, their sum is \( 180^\circ \), giving:
   \[
   148^\circ + (3x - y)^\circ = 180^\circ
   \]
   Simplifying:
   \[
   3x - y = 180^\circ - 148^\circ = 32^\circ \quad \text{(Equation 2)}
   \]

### Now, solve the system of equations:

1. From Equation 1: \( 3x + y = 52^\circ \)
2. From Equation 2: \( 3x - y = 32^\circ \)

We can solve these two equations simultaneously by adding them together to eliminate \( y \):

\[
(3x + y) + (3x - y) = 52^\circ + 32^\circ
\]
\[
6x = 84^\circ
\]
\[
x = \frac{84^\circ}{6} = 14^\circ
\]

Now that we have \( x = 14^\circ \), substitute this value into either equation to find \( y \). Using Equation 1:

\[
3(14^\circ) + y = 52^\circ
\]
\[
42^\circ + y = 52^\circ
\]
\[
y = 52^\circ - 42^\circ = 10^\circ
\]

### Final Answer:
- \( x = 14^\circ \)
- \( y = 10^\circ \)

Would you like further details or have any questions? 

Here are 5 related questions you might explore:

1. What are consecutive interior angles, and why do they add up to 180°?
2. How would the solution change if the transversal angle measurements were altered?
3. Can corresponding angles be used to find \( x \) and \( y \) in a similar problem?
4. What are the other angle pairs that are congruent when a transversal crosses parallel lines?
5. How do you prove that two lines are parallel using angle relationships?

**Tip:** In problems involving parallel lines, always consider the angle relationships: corresponding, alternate interior, alternate exterior, and consecutive interior angles.

Find the value of x and y for the given angle relationships in the diagram where the angles formed by parallel lines and transversals are: 128°, (3x + y)°, (3x - y)°, and 148°.

In this problem, solve for x and y by applying the consecutive interior angles theorem. Explore the relationships formed by parallel lines and transversals, using algebraic methods to solve the system of equations. This problem is suitable for Grades 8-10 students.

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