Two unequal angles of a parallelogram are in the ratio `2: 3.` Find all its angles in degrees.

Options

A. `72^@` and `108^@`
B. `70^@` and `110^@`
C. `60^@` and `120^@`
D. none of these

Answer

To find the angles of a parallelogram where two unequal angles are in the ratio of 2:3, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of a Parallelogram**:
- In a parallelogram, opposite angles are equal.
- The sum of all angles in a parallelogram is 360 degrees.

2. **Set Up the Ratio**:
- Let the two unequal angles be represented as \( x \) and \( y \).
- According to the problem, the ratio of the angles is given as \( 2:3 \).
- We can express this as:
\[ x = 2r \quad \text{and} \quad y = 3r \]
where \( r \) is a common multiplier.

3. **Write the Equation for the Sum of Angles**:
- Since opposite angles are equal, the total sum of angles can be expressed as:
\[ x + y + x + y = 360^\circ \]
- This simplifies to:
\[ 2x + 2y = 360^\circ \]
- Dividing the entire equation by 2 gives:
\[ x + y = 180^\circ \]

4. **Substitute the Values of \( x \) and \( y \)**:
- Substitute \( x = 2r \) and \( y = 3r \) into the equation:
\[ 2r + 3r = 180^\circ \]
- This simplifies to:
\[ 5r = 180^\circ \]

5. **Solve for \( r \)**:
- Divide both sides by 5:
\[ r = \frac{180^\circ}{5} = 36^\circ \]

6. **Find the Values of \( x \) and \( y \)**:
- Now substitute \( r \) back into the expressions for \( x \) and \( y \):
\[ x = 2r = 2 \times 36^\circ = 72^\circ \]
\[ y = 3r = 3 \times 36^\circ = 108^\circ \]

7. **List All Angles of the Parallelogram**:
- The angles of the parallelogram are:
- \( 72^\circ \) (two angles)
- \( 108^\circ \) (two angles)

### Final Answer:
The angles of the parallelogram are \( 72^\circ, 108^\circ, 72^\circ, \) and \( 108^\circ \).

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