To solve for angle $\angle 1$, let's analyze the information provided.

Step-by-Step Solution

1. Identify Parallel Lines and Transversals:

   • Lines $a$ and $b$ are parallel, and lines $c$ and $d$ appear to intersect them, acting as transversals.

2. Set Up Angle Relationships:

   • Given that angles formed by parallel lines and transversals are related, we can use corresponding angles, alternate interior angles, or linear pair relationships.

3. Using the Given Angles:

   • We have two expressions:
     • $3x^\circ$, which is one of the angles created by the transversal at line $a$,
     • $(5x - 45)^\circ$, which is the corresponding angle on line $b$.

4. Equate Corresponding Angles:

   • Since $a \parallel b$, corresponding angles are equal: $3x = 5x - 45$

5. Solve for $x$:

   • Rearranging the equation: $3x - 5x = -45$ $-2x = -45$ $x = 22.5$

6. Find $\angle 1$:

   • Substitute $x = 22.5$ back into $3x$: $3x = 3 \times 22.5 = 67.5^\circ$

Conclusion

The measure of $\angle 1$ is $67.5^\circ$.

Answer:

Option B: The measure of $\angle 1$ is $67.5^\circ$.

Would you like further details or have any questions?

Further Questions to Explore:

1. What are the properties of corresponding and alternate interior angles in parallel lines?
2. How would the solution change if $\angle 1$ were an alternate interior angle?
3. Can you solve for any other angles in the figure based on the value of $x$?
4. What would happen if $a$ and $b$ were not parallel?
5. How are transversals used in geometry to determine unknown angles?

Tip:

Always check if lines are parallel before using angle relationships like corresponding or alternate interior angles.