Clock angle — What is the (smaller) angle between the minute and hour hands at exactly 2:45?

Introduction / Context:
We compute the smaller angle between the hands at 2:45 using standard formulas.

Given Data / Assumptions:

• At 2:45 → h = 2, m = 45.
• Hour angle = 30h + 0.5m; minute angle = 6m.

Concept / Approach:
Angle difference = |(30h + 0.5m) − 6m|; report the smaller angle.

Step-by-Step Solution:
Hour angle = 30*2 + 0.5*45 = 60 + 22.5 = 82.5.Minute angle = 6*45 = 270.Difference = |82.5 − 270| = 187.5 degrees.Smaller angle = 360 − 187.5 = 172.5 degrees.

Verification / Alternative check:
Using the compact formula |30h − 5.5m| also yields |60 − 247.5| = 187.5 → smaller 172.5.

Why Other Options Are Wrong:
180°, 190°, 175° are nearby round values but not exact; correct is 172.5°.

Common Pitfalls:
Reporting the reflex angle (187.5°) instead of the smaller one, or rounding to 180°.

Final Answer:
172.5°, which is not listed; hence “None of these.”