Of the three angles of a triangle, one is twice the smallest and another one is thrice the smallest. Find the angles.

Answer

To find the angles of the triangle, we can follow these steps:

### Step 1: Define the smallest angle
Let the smallest angle of the triangle be \( x \).

### Step 2: Express the other angles in terms of \( x \)
According to the problem:
- One angle (let's call it angle A) is twice the smallest angle:
\[ A = 2x \]
- Another angle (let's call it angle B) is thrice the smallest angle:
\[ B = 3x \]

### Step 3: Write the equation for the sum of angles in a triangle
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can write the equation:
\[ A + B + C = 180^\circ \]
Substituting the values of A, B, and C:
\[ 2x + 3x + x = 180^\circ \]

### Step 4: Combine like terms
Combine the terms on the left side:
\[ 6x = 180^\circ \]

### Step 5: Solve for \( x \)
To find \( x \), divide both sides by 6:
\[ x = \frac{180^\circ}{6} = 30^\circ \]

### Step 6: Find the angles A, B, and C
Now that we have \( x \):
- Angle C (the smallest angle) is:
\[ C = x = 30^\circ \]
- Angle A is:
\[ A = 2x = 2 \times 30^\circ = 60^\circ \]
- Angle B is:
\[ B = 3x = 3 \times 30^\circ = 90^\circ \]

### Final Answer
The angles of the triangle are:
- Angle A = \( 60^\circ \)
- Angle B = \( 90^\circ \)
- Angle C = \( 30^\circ \)

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