If two acute angles of a right angled triangle are in the ratio 2 : 3 , find the measure of all angles of the triangle.

Answer

To solve the problem, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know that in a right-angled triangle, one angle is always 90 degrees. The other two angles are acute angles and are given to be in the ratio of 2:3.

2. **Assign Variables**: Let the two acute angles be represented as \(2x\) and \(3x\) based on the given ratio.

3. **Set Up the Equation**: Since the sum of all angles in a triangle is 180 degrees, we can write the equation:
\[ 2x + 3x + 90 = 180 \]

4. **Combine Like Terms**: Simplify the equation:
\[ 5x + 90 = 180 \]

5. **Isolate \(x\)**: Subtract 90 from both sides:
\[ 5x = 180 - 90 \]
\[ 5x = 90 \]

6. **Solve for \(x\)**: Divide both sides by 5:
\[ x = \frac{90}{5} = 18 \]

7. **Find the Angles**: Now substitute \(x\) back into the expressions for the angles:
- First angle: \(2x = 2 \times 18 = 36\) degrees
- Second angle: \(3x = 3 \times 18 = 54\) degrees

8. **List All Angles**: The angles of the triangle are:
- First angle: \(36\) degrees
- Second angle: \(54\) degrees
- Third angle: \(90\) degrees (right angle)

### Final Answer:
The measures of all angles of the triangle are \(36\) degrees, \(54\) degrees, and \(90\) degrees.

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