The angles of a cyclic quadrilateral are in the ratio 1:2:3:4. What is the measure of the smallest angle?

Question

The angles of a cyclic quadrilateral are in the ratio 1:2:3:4. What is the measure of the smallest angle?

1. A. 36° (Correct answer)
2. B. 72°
3. C. 108°
4. D. 144°

Correct Answer

Option A — 36°

Detailed Solution & Explanation

The correct answer is 36°.

Key Points

• The angles of any quadrilateral sum to 360°.
• With the angles in $1 : 2 : 3 : 4$ there are $1 + 2 + 3 + 4 = 10$ parts, so one part is $360 \div 10 = 36°$.
• The smallest angle is 1 part = 36°.

Additional Information

• In a cyclic quadrilateral opposite angles are supplementary. Check: $36 + 108 = 144$ and $72 + 144 = 216$ — so this set is not in fact cyclic, but the angle-sum property still fixes the answer.
• The angles work out as 36°, 72°, 108° and 144°, summing to 360° ✓
• The general angle sum for an n-sided polygon is $(n - 2) \times 180°$, which gives 360° for a quadrilateral.

प्रश्न (हिन्दी में)

एक चक्रीय चतुर्भुज के कोणों का अनुपात 1:2:3:4 है। सबसे छोटे कोण का माप क्या है?

1. A. 36°
2. B. 72°
3. C. 108°
4. D. 144°