A girl is initially standing facing towards the south. She first turns 135 degrees in the anticlockwise direction, and then again takes a 180 degree turn in the anticlockwise direction. After performing these two rotations, in which direction is she finally facing?

Introduction / Context:
This question is purely about orientation and rotation in terms of degrees. It asks for the final facing direction of a girl after she turns first 135 degrees anticlockwise from facing south and then another 180 degrees anticlockwise. Correctly handling angles larger than 90 degrees and remembering the relative positions of the eight principal directions is essential.

Given Data / Assumptions:

• The girl initially faces south.
• She turns 135 degrees anticlockwise.
• From that new direction, she again turns 180 degrees anticlockwise.
• We use the standard compass with North, East, South, and West, and the four diagonal directions North-east, South-east, South-west, and North-west.
• All rotations are in the horizontal plane.

Concept / Approach:
We can represent directions on a circle: East at 0 degrees, North at 90 degrees, West at 180 degrees, and South at 270 degrees. Anticlockwise movement corresponds to increasing angle, while clockwise movement corresponds to decreasing angle. Each 45 degree increment gives an intercardinal direction. By converting rotations into final angles, we can relate them back to the nearest named direction.

Step-by-Step Solution:
Step 1: Represent the initial direction. Facing south corresponds to 270 degrees on the directional circle if East is taken as 0 degrees. Step 2: First rotation: she turns 135 degrees anticlockwise. Adding 135 degrees to 270 degrees gives 270 + 135 = 405 degrees. Step 3: Since angles wrap around every 360 degrees, 405 degrees is equivalent to 405 − 360 = 45 degrees. The direction at 45 degrees is North-east. Step 4: Second rotation: now from North-east (45 degrees), she turns a further 180 degrees anticlockwise. Adding 180 degrees gives 45 + 180 = 225 degrees. Step 5: The direction of 225 degrees lies exactly halfway between South (270 degrees) and West (180 degrees), so it is the South-west direction.

Verification / Alternative check:
You can check using quadrant thinking. From South, a 90 degree anticlockwise turn leads to West, and an additional 45 degrees leads to North-west. However, we need 135 degrees, which carries her to North-east due to wrapping around the circle past West and North. Rotating another 180 degrees from North-east places her diagonally opposite, that is, pointing to South-west, which is consistent with the angle based method.

Why Other Options Are Wrong:
South would be correct only if the final net rotation were a multiple of 360 degrees from the initial direction, which it is not. South-east and West correspond to angles of 135 degrees and 180 degrees from East respectively, which do not match the calculated 225 degrees final angle. Thus, they do not represent the correct final orientation.

Common Pitfalls:
Mistakes often come from treating 135 degrees as 90 degrees, from miscounting anticlockwise versus clockwise, or from forgetting to wrap angles greater than 360 degrees. Another pitfall is confusing North-east with North-west when working with diagonal directions. Drawing a circle with marked angles greatly helps to avoid such confusion.

Final Answer:
After both anticlockwise turns, the girl is facing the South-west direction.