All the four angles of a quadrilateral are in G.P. with common ratio r`(r in N)` . Any two angles are acute and other two angles are obtuse, then the measure of the smallest angle of the quadrilateral is :

Options

A. `18^@`
B. `32^@`
C. `24^@`
D. `36^@`

Answer

To solve the problem, we need to find the smallest angle of a quadrilateral where all four angles are in a geometric progression (G.P.) with a common ratio \( r \) (where \( r \) is a natural number). We also know that two angles are acute and two angles are obtuse.

### Step-by-Step Solution:

1. **Define the Angles**:
Let the four angles of the quadrilateral be represented as:
- \( a \) (acute)
- \( ar \) (acute)
- \( ar^2 \) (obtuse)
- \( ar^3 \) (obtuse)

Here, \( a \) is the first angle and \( r \) is the common ratio.

2. **Sum of Angles in a Quadrilateral**:
The sum of the angles in any quadrilateral is \( 360^\circ \). Therefore, we can write the equation:
\[ a + ar + ar^2 + ar^3 = 360 \]

3. **Factor Out \( a \)**:
We can factor out \( a \) from the left side:
\[ a(1 + r + r^2 + r^3) = 360 \]

4. **Solve for \( a \)**:
Rearranging gives:
\[ a = \frac{360}{1 + r + r^2 + r^3} \]

5. **Determine Conditions for Angles**:
Since two angles are acute, we need:
- \( a < 90 \)
- \( ar < 90 \)

For the obtuse angles, we need:
- \( ar^2 > 90 \)
- \( ar^3 > 90 \)

6. **Substituting \( a \)**:
Substitute \( a \) into the inequalities:
- \( \frac{360}{1 + r + r^2 + r^3} < 90 \)
- \( \frac{360r}{1 + r + r^2 + r^3} < 90 \)

This simplifies to:
- \( 1 + r + r^2 + r^3 > 4 \) (for the first inequality)
- \( 1 + r + r^2 + r^3 > 4r \) (for the second inequality)

7. **Finding Values of \( r \)**:
We can test natural numbers for \( r \):
- For \( r = 1 \):
- \( 1 + 1 + 1 + 1 = 4 \) (not valid)
- For \( r = 2 \):
- \( 1 + 2 + 4 + 8 = 15 \) (valid)
- For \( r = 3 \):
- \( 1 + 3 + 9 + 27 = 40 \) (valid)

8. **Calculating \( a \) for \( r = 2 \)**:
Substitute \( r = 2 \) into the equation for \( a \):
\[ a = \frac{360}{1 + 2 + 4 + 8} = \frac{360}{15} = 24 \]

9. **Finding Other Angles**:
- \( ar = 48 \)
- \( ar^2 = 96 \)
- \( ar^3 = 192 \)

10. **Conclusion**:
The smallest angle of the quadrilateral is \( 24^\circ \).

### Final Answer:
The measure of the smallest angle of the quadrilateral is \( 24^\circ \).