Angles of a quadrilateral are in the ratio 3:4:5:8. The smallest angle is

Options

A. `54^(@)`
B. `40^(@)`
C. `36^(@)`
D. `18^(@)`

Answer

To find the smallest angle of a quadrilateral where the angles are in the ratio 3:4:5:8, we can follow these steps:

### Step 1: Understand the properties of a quadrilateral
The sum of all angles in a quadrilateral is always 360 degrees.

### Step 2: Assign variables based on the given ratio
Let the angles be represented as:
- Angle A = 3x
- Angle B = 4x
- Angle C = 5x
- Angle D = 8x

### Step 3: Set up the equation based on the sum of angles
According to the property of quadrilaterals:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360 \]
Substituting the values we assigned:
\[ 3x + 4x + 5x + 8x = 360 \]

### Step 4: Combine like terms
Combine the terms on the left side:
\[ (3 + 4 + 5 + 8)x = 360 \]
\[ 20x = 360 \]

### Step 5: Solve for x
To find the value of x, divide both sides by 20:
\[ x = \frac{360}{20} \]
\[ x = 18 \]

### Step 6: Calculate each angle
Now substitute the value of x back into the expressions for each angle:
- Angle A = \( 3x = 3 \times 18 = 54^\circ \)
- Angle B = \( 4x = 4 \times 18 = 72^\circ \)
- Angle C = \( 5x = 5 \times 18 = 90^\circ \)
- Angle D = \( 8x = 8 \times 18 = 144^\circ \)

### Step 7: Identify the smallest angle
Among the calculated angles:
- Angle A = 54°
- Angle B = 72°
- Angle C = 90°
- Angle D = 144°

The smallest angle is **54°**.

### Final Answer:
The smallest angle in the quadrilateral is **54°**.

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