https://file.aiquickdraw.com/mathbot/file/1730147103298m5ap8j1c.jpg

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?

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#### 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.

Solve for the measure of angle 1 using properties of parallel lines and transversals. Learn how to equate corresponding angles to find the solution, including a step-by-step solution. Suitable for students in Grades 8-10.

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