The four angles of a quadrilateral are in the ratio `2:3:5:8`. Find the angles.

Options

A. `40^(@), 60^(@), 120^(@) and 140^(@)`.
B. `80^(@), 60^(@), 80^(@) and 160^(@)`.
C. `40^(@), 60^(@), 100^(@) and 160^(@)`.
D. `40^(@), 70^(@), 100^(@) and 160^(@)`.

Answer

To find the angles of a quadrilateral given in the ratio 2:3:5:8, we can follow these steps:

### Step 1: Assign Variables Based on the Ratio
Let the angles of the quadrilateral be represented as:
- Angle A = 2x
- Angle B = 3x
- Angle C = 5x
- Angle D = 8x

### Step 2: Use the Property of Quadrilaterals
The sum of the angles in any quadrilateral is always 360 degrees. Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360^\circ \]
Substituting the expressions for the angles we have:
\[ 2x + 3x + 5x + 8x = 360^\circ \]

### Step 3: Combine Like Terms
Now combine the terms on the left side:
\[ (2x + 3x + 5x + 8x) = 18x \]
So, we have:
\[ 18x = 360^\circ \]

### Step 4: Solve for x
To find the value of x, divide both sides of the equation by 18:
\[ x = \frac{360^\circ}{18} \]
Calculating this gives:
\[ x = 20^\circ \]

### Step 5: Calculate Each Angle
Now that we have the value of x, we can find each angle:
- Angle A = \( 2x = 2 \times 20^\circ = 40^\circ \)
- Angle B = \( 3x = 3 \times 20^\circ = 60^\circ \)
- Angle C = \( 5x = 5 \times 20^\circ = 100^\circ \)
- Angle D = \( 8x = 8 \times 20^\circ = 160^\circ \)

### Step 6: State the Final Angles
Thus, the angles of the quadrilateral are:
- Angle A = 40 degrees
- Angle B = 60 degrees
- Angle C = 100 degrees
- Angle D = 160 degrees

### Summary of Angles
The angles of the quadrilateral are:
- 40°, 60°, 100°, and 160°.

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