1. The problem asks: In which quadrant does the terminal side of a $-45^\circ$ angle in standard position lie? 2. Recall that angles in standard position start from the positive x-axis and rotate counterclockwise for positive angles and clockwise for negative angles. 3. A $-45^\circ$ angle means rotating $45^\circ$ clockwise from the positive x-axis. 4. The positive x-axis is the boundary between Quadrant I and Quadrant IV. 5. Rotating $45^\circ$ clockwise places the terminal side in Quadrant IV. 6. Therefore, the terminal side of a $-45^\circ$ angle lies in Quadrant IV.