If the interior angles of a five sided polygon are in the ratio of 2:3:3:5:5, then the measure of the smallest angles is
यदि पांच भुजाओं वाले किसी बहुभुज के अंत:कोण 2:3:3:5:5 के अनुपात में हैं, तो सबसे छोटे कोण का माप है

Options

A. `20^(@)`
B. `30^(@)`
C. `60^(@)`
D. `90^(@)`

Explanation

To find the measure of the smallest angle in a five-sided polygon (pentagon) where the interior angles are in the ratio of 2:3:3:5:5, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Ratio**:
The angles of the polygon are given in the ratio 2:3:3:5:5. We can express the angles in terms of a variable \( x \):
- Let the angles be:
- Angle A = \( 2x \)
- Angle B = \( 3x \)
- Angle C = \( 3x \)
- Angle D = \( 5x \)
- Angle E = \( 5x \)

2. **Sum of Interior Angles of a Pentagon**:
The formula for the sum of the interior angles of a polygon is given by:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides. For a pentagon, \( n = 5 \):
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]

3. **Setting Up the Equation**:
Now, we can set up the equation for the sum of the angles:
\[ 2x + 3x + 3x + 5x + 5x = 540^\circ \]

4. **Simplifying the Equation**:
Combine like terms:
\[ (2x + 3x + 3x + 5x + 5x) = 18x \]
Therefore, we have:
\[ 18x = 540^\circ \]

5. **Solving for \( x \)**:
Divide both sides by 18 to find \( x \):
\[ x = \frac{540^\circ}{18} = 30^\circ \]

6. **Finding the Smallest Angle**:
The smallest angle is \( 2x \):
\[ \text{Smallest angle} = 2x = 2 \times 30^\circ = 60^\circ \]

### Final Answer:
The measure of the smallest angle is \( 60^\circ \).

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