The ratio of angles of a pentagon is 1 : 2 : 3 : 5 : 9, the value of its smallest angle is :

Question

Accepted Answer

To determine the smallest angle of a pentagon with angles in the ratio 1:2:3:5:9, we first need to understand the properties of a pentagon and how angle ratios work.

1. **Sum of Interior Angles of a Pentagon:**
   A pentagon has five sides, and the sum of its interior angles can be calculated using the formula:
   $$
   \text{Sum of interior angles} = (n - 2) \times 180^\circ
   $$
   where $ n $ is the number of sides. For a pentagon, $ n = 5 $:
   $$
   \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
   $$

2. **Using the Ratio to Find Individual Angles:**
   The angles are in the ratio 1:2:3:5:9. Let the common multiple be $ x $. Then the angles can be expressed as:
   - Smallest angle = $ 1x $
   - Second angle = $ 2x $
   - Third angle = $ 3x $
   - Fourth angle = $ 5x $
   - Fifth angle = $ 9x $

The sum of these angles should equal the total sum of the interior angles of the pentagon:
   $$
   1x + 2x + 3x + 5x + 9x = 540^\circ
   $$
   $$
   20x = 540^\circ
   $$
   $$
   x = \frac{540^\circ}{20} = 27^\circ
   $$

3. **Calculating the Smallest Angle:**
   The smallest angle is $ 1x = 27^\circ $.

4. **Explanation of Why Option 4 is Correct:**
   Based on the calculations above, the smallest angle of the pentagon is $ 27^\circ $, which corresponds to Option 4. This confirms that Option 4 is indeed the correct answer.

5. **Explanation of Why Other Options are Incorrect:**
   - **Option 1 (22°):** This value is incorrect because, based on the ratio and the sum of the angles, the smallest angle must be $ 27^\circ $.
   - **Option 2 (34°):** This value is incorrect as it does not satisfy the equation derived from the ratio and the sum of the angles.
   - **Option 3 (45°):** This value is incorrect because it exceeds the calculated smallest angle of $ 27^\circ $.

In conclusion, the correct answer is Option 4, $ 27^\circ $, as it accurately reflects the smallest angle of the pentagon based on the given ratio and the properties of a pentagon's interior angles.