Two angles of a pentagon are in the ratio 2:3. The others are 60o each. Calculate the smaller of the two angles

Analyze the question: A pentagon has 5 angles. Two of them are in the ratio 2:3, and the other three are 60° each. We need the smaller of the two angles.

Concept: The sum of interior angles of a polygon with n sides is (n − 2) × 180°.

For a pentagon (n = 5): Sum = (5 − 2) × 180° = 3 × 180° = 540°

Solution:

1. The three equal angles total: 3 × 60° = 180°
2. The remaining two angles total: 540° − 180° = 360°
3. Share 360° in the ratio 2:3 (total 5 parts): • One part = 360° ÷ 5 = 72° • Smaller angle = 2 × 72° = 144° • Larger angle = 3 × 72° = 216°

Check: 144° + 216° + 60° + 60° + 60° = 540° ✓

(The larger angle is 216°, which is a reflex angle, so the pentagon is non-convex. This is allowed, and the question only asks for the smaller angle.)

Answer: The smaller angle is 144°, which is option D.

Tip: Always start polygon angle problems with (n − 2) × 180°. Subtract the known angles, then divide the remainder by the total ratio parts. Option A (72°) is a trap: it is just one ratio part, so remember to multiply by 2.