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Let's analyze both parts of the problem to determine whether lines \( a \) and \( b \) can be proven parallel.

### Part a)
In the first diagram, we see that lines \( a \) and \( b \) are crossed by a horizontal transversal, forming a box symbol at the intersection. This symbol indicates a 90-degree angle at all four corners of the intersection. If two lines intersect a transversal at 90-degree angles, the lines are perpendicular to the transversal, which is sufficient to prove that lines \( a \) and \( b \) are parallel.

**Conclusion for Part a**: Yes, it is possible to prove that lines \( a \) and \( b \) are parallel because both lines are perpendicular to the transversal, meaning they have the same direction relative to the transversal and do not intersect.

### Part b)
In the second diagram, lines \( a \) and \( b \) are crossed by a transversal, forming several angles. To prove parallelism, we can use angle relationships. Here are the given angles:

- The alternate interior angles are \( 66^\circ \) and \( 66^\circ \). 
- The corresponding angle is \( 48^\circ \).
- The supplementary angle is \( 114^\circ \).

If alternate interior angles are equal, it is a property of parallel lines. Since both alternate interior angles are \( 66^\circ \), this is sufficient to prove that lines \( a \) and \( b \) are parallel.

**Conclusion for Part b**: Yes, lines \( a \) and \( b \) can be proven parallel because the alternate interior angles are equal.

### Summary:
- Part a: Lines \( a \) and \( b \) are parallel because they are both perpendicular to the transversal.
- Part b: Lines \( a \) and \( b \) are parallel because the alternate interior angles formed by the transversal are equal.

Would you like further details or explanations on any of these concepts? 

Here are five related questions you might find helpful:
1. How do perpendicular lines relate to proving parallelism?
2. What are alternate interior angles, and why do they prove parallelism?
3. Can corresponding angles also prove parallelism?
4. What other angle relationships can be used to prove that lines are parallel?
5. How can supplementary angles help in identifying parallel lines?

**Tip**: When working with parallel lines and transversals, always look for angle relationships such as alternate interior, corresponding, or supplementary angles to determine parallelism.

Is it possible to prove lines a and b are parallel in the two diagrams provided?

Learn how to determine whether lines a and b are parallel using transversal properties, 90-degree angles, and alternate interior angles. This problem involves basic geometry concepts suitable for students in Grades 7-9.

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