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.