One angle of a pentagon is 140° and the remaining angles are in the ratio 1 : 2 : 3 : 4. The largest angle of the pentagon is equal to :

Question

Accepted Answer

To determine the largest angle of the pentagon, we start by understanding the given information and applying the properties of polygons.

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. **Given Information:**
   One angle of the pentagon is $ 140^\circ $. The remaining four angles are in the ratio 1 : 2 : 3 : 4.

3. **Calculating the Remaining Angles:**
   Let the four angles be $ x, 2x, 3x, $ and $ 4x $. According to the problem, the sum of these angles plus the given angle should equal the total sum of the pentagon's interior angles:
   $$
   140^\circ + x + 2x + 3x + 4x = 540^\circ
   $$
   Simplifying the equation:
   $$
   140^\circ + 10x = 540^\circ
   $$
   $$
   10x = 540^\circ - 140^\circ = 400^\circ
   $$
   $$
   x = \frac{400^\circ}{10} = 40^\circ
   $$

4. **Finding the Largest Angle:**
   The angles are $ x = 40^\circ, 2x = 80^\circ, 3x = 120^\circ, $ and $ 4x = 160^\circ $. The largest angle among these is $ 160^\circ $.

5. **Conclusion:**
   Therefore, the largest angle of the pentagon is $ 160^\circ $, which corresponds to Option 2.

**Explanation of Incorrect Options:**

- **Option 1 (168°):** This angle is not possible given the calculated values of $ x, 2x, 3x, $ and $ 4x $. The largest angle calculated is $ 160^\circ $, not $ 168^\circ $.

- **Option 3 (150°):** This angle is also not possible as it does not match any of the calculated angles based on the given ratio.

- **Option 4 (140°):** This is the angle that was already provided in the problem, not the largest angle calculated from the ratio.

Thus, the correct answer is Option 2, $ 160^\circ $, as it logically follows from the given conditions and calculations.