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

Answer

To find all 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. **Understanding the Properties of a Parallelogram**:
In a parallelogram, opposite angles are equal, and the sum of adjacent angles is supplementary (i.e., they add up to 180 degrees).

2. **Let the Angles be Represented in Terms of a Variable**:
Let the two unequal angles be represented as:
- Angle A = 2k
- Angle D = 3k
Here, k is a common multiplier.

3. **Set Up the Equation for Supplementary Angles**:
Since angle A and angle D are adjacent angles, we can write the equation:
\[ \text{Angle A} + \text{Angle D} = 180^\circ \]
Substituting the expressions for angle A and angle D, we get:
\[ 2k + 3k = 180^\circ \]

4. **Combine Like Terms**:
Combine the terms on the left side:
\[ 5k = 180^\circ \]

5. **Solve for k**:
Divide both sides by 5 to find k:
\[ k = \frac{180^\circ}{5} = 36^\circ \]

6. **Calculate the Angles**:
Now that we have the value of k, we can find the angles:
- Angle A = 2k = 2 × 36° = 72°
- Angle D = 3k = 3 × 36° = 108°

7. **Determine the Remaining Angles**:
Since opposite angles in a parallelogram are equal:
- Angle B = Angle D = 108°
- Angle C = Angle A = 72°

8. **Final Angles of the Parallelogram**:
Therefore, the angles of the parallelogram are:
- Angle A = 72°
- Angle B = 108°
- Angle C = 72°
- Angle D = 108°

### Summary of Angles:
- Angle A = 72°
- Angle B = 108°
- Angle C = 72°
- Angle D = 108°