Let's analyze each problem and find the value of $x$ by using the concept of corresponding angles and supplementary angles.

1. Problem 1: $x + 34^\circ = 146^\circ$ (since they are supplementary)

   • Solution: $x = 146^\circ - 34^\circ = 112^\circ$

2. Problem 2: $x = 93^\circ$ (corresponding angles)

   • Solution: $x = 93^\circ$

3. Problem 3: $x + 75^\circ = 105^\circ$ (supplementary)

   • Solution: $x = 105^\circ - 75^\circ = 30^\circ$

4. Problem 4: $x = 69^\circ$ (corresponding angles)

   • Solution: $x = 69^\circ$

5. Problem 5: $x + 30^\circ = 150^\circ$ (supplementary)

   • Solution: $x = 150^\circ - 30^\circ = 120^\circ$

6. Problem 6: $x = 128^\circ$ (corresponding angles)

   • Solution: $x = 128^\circ$

7. Problem 7: $x + 49^\circ = 131^\circ$ (supplementary)

   • Solution: $x = 131^\circ - 49^\circ = 82^\circ$

8. Problem 8: $x = 103^\circ$ (corresponding angles)

   • Solution: $x = 103^\circ$

9. Problem 9: $x + 66^\circ = 114^\circ$ (supplementary)

   • Solution: $x = 114^\circ - 66^\circ = 48^\circ$

Here are the solutions:

1. $x = 112^\circ$
2. $x = 93^\circ$
3. $x = 30^\circ$
4. $x = 69^\circ$
5. $x = 120^\circ$
6. $x = 128^\circ$
7. $x = 82^\circ$
8. $x = 103^\circ$
9. $x = 48^\circ$

Would you like a detailed explanation for any specific problem, or have any questions?

Here are some related questions to expand understanding:

1. What are corresponding angles and how are they identified?
2. How do we use the concept of supplementary angles to solve for unknown angles?
3. Can corresponding angles be unequal in any scenario involving parallel lines and a transversal?
4. What is the difference between alternate interior and alternate exterior angles?
5. How can we verify if two lines are parallel by observing angles formed with a transversal?

Tip: In a pair of parallel lines cut by a transversal, corresponding angles are always equal, while consecutive interior angles are supplementary.