1. The problem is to find in which quadrant the angle 135 degrees lies and to find a co-terminal angle for it. 2. Angles are measured from the positive x-axis, counterclockwise. The four quadrants are: - Quadrant I: $0^\circ$ to $90^\circ$ - Quadrant II: $90^\circ$ to $180^\circ$ - Quadrant III: $180^\circ$ to $270^\circ$ - Quadrant IV: $270^\circ$ to $360^\circ$ 3. Since $135^\circ$ is between $90^\circ$ and $180^\circ$, it lies in Quadrant II. 4. Co-terminal angles differ by full rotations of $360^\circ$. The formula for co-terminal angles is: $$\theta_{coterminal} = \theta \pm 360^\circ k$$ where $k$ is any integer. 5. For $135^\circ$, one co-terminal angle can be found by subtracting $360^\circ$: $$135^\circ - 360^\circ = -225^\circ$$ 6. Another co-terminal angle can be found by adding $360^\circ$: $$135^\circ + 360^\circ = 495^\circ$$ Therefore, $135^\circ$ lies in Quadrant II, and examples of co-terminal angles are $-225^\circ$ and $495^\circ$.