Q.Draw the angles of the following measures and determine their quadrants.

Step 1: Sketch each angle in standard position (initial arm along the positive X-axis, rotated anticlockwise for a positive measure and clockwise for a negative measure), then reduce it modulo $360°$ to a value in $[0°,360°)$ to identify the quadrant of the terminal ray: I is $0°$–$90°$, II is $90°$–$180°$, III is $180°$–$270°$, IV is $270°$–$360°$.

Step 2: Reduce each angle.

i) $-140° + 360° = 220°$ → Quadrant III.

ii) $250°$ is already in range → Quadrant III.

iii) $420° - 360° = 60°$ → Quadrant I.

iv) $750° - 720° = 30°$ → Quadrant I.

v) $945° - 720° = 225°$ → Quadrant III.

vi) $1120° - 1080° = 40°$ → Quadrant I.

vii) $-80° + 360° = 280°$ → Quadrant IV.

viii) $-330° + 360° = 30°$ → Quadrant I.

ix) $-500° + 720° = 220°$ → Quadrant III.

x) $-820° + 1080° = 260°$ → Quadrant III.

Reduce each angle modulo 360° to the range [0°,360°) and compare against the quadrant boundaries 90°, 180°, 270°.

• Subtracting only one multiple of 360° when a larger angle needs two or three subtracted.
• Forgetting that a negative angle is reduced by adding (not subtracting) 360° until it lands in [0°,360°).
• Misreading a boundary angle (exactly 90°, 180°, 270°, 360°) as belonging to a quadrant rather than being a quadrantal angle.