In a `DeltaABC, angleA=2 angleB=3 angleC,` find each of the triangle.

Answer

To solve the problem, we need to find the angles of triangle ABC given the relationships between the angles: \( \angle A = 2 \angle B \) and \( \angle A = 3 \angle C \).

### Step-by-Step Solution:

1. **Set Up the Relationships**:
We know from the problem statement that:
- \( \angle A = 2 \angle B \) (1)
- \( \angle A = 3 \angle C \) (2)

2. **Express Angles in Terms of Angle B**:
From equation (1), we can express \( \angle A \) in terms of \( \angle B \):
\[ \angle A = 2B \]
From equation (2), we can express \( \angle C \) in terms of \( \angle B \):
\[ \angle C = \frac{2}{3} \angle B \]

3. **Use the Angle Sum Property**:
The sum of the angles in a triangle is always 180 degrees:
\[ \angle A + \angle B + \angle C = 180 \]
Substituting the expressions for \( \angle A \) and \( \angle C \):
\[ 2B + B + \frac{2}{3}B = 180 \]

4. **Combine Like Terms**:
Combine the terms on the left side:
\[ 3B + \frac{2}{3}B = 180 \]
To combine \( 3B \) and \( \frac{2}{3}B \), we convert \( 3B \) into a fraction:
\[ 3B = \frac{9}{3}B \]
Now we can add:
\[ \frac{9}{3}B + \frac{2}{3}B = \frac{11}{3}B \]
So we have:
\[ \frac{11}{3}B = 180 \]

5. **Solve for Angle B**:
Multiply both sides by \( \frac{3}{11} \):
\[ B = 180 \times \frac{3}{11} = \frac{540}{11} \]
Converting this to a mixed number:
\[ B = 49 \frac{1}{11} \text{ degrees} \]

6. **Find Angle A**:
Using \( \angle A = 2B \):
\[ A = 2 \times \frac{540}{11} = \frac{1080}{11} \]
Converting to a mixed number:
\[ A = 98 \frac{2}{11} \text{ degrees} \]

7. **Find Angle C**:
Using \( \angle C = \frac{2}{3}B \):
\[ C = \frac{2}{3} \times \frac{540}{11} = \frac{360}{11} \]
Converting to a mixed number:
\[ C = 32 \frac{8}{11} \text{ degrees} \]

### Final Angles:
- \( \angle A = 98 \frac{2}{11} \) degrees
- \( \angle B = 49 \frac{1}{11} \) degrees
- \( \angle C = 32 \frac{8}{11} \) degrees