The largest angle of a triangle is twice the sum of the other two. The smallest is one-fourth of the largest. Determine all the angles in degrees.

Answer

To solve the problem, we will define the angles of the triangle and set up equations based on the information provided.

### Step 1: Define the angles
Let:
- \( x \) = the smallest angle
- \( y \) = the second angle
- \( z \) = the largest angle

### Step 2: Set up the equations
From the problem statement, we have two key pieces of information:
1. The largest angle \( z \) is twice the sum of the other two angles:
\[ z = 2(x + y) \quad \text{(Equation 1)} \]
2. The smallest angle \( x \) is one-fourth of the largest angle \( z \):
\[ x = \frac{1}{4}z \quad \text{(Equation 2)} \]

### Step 3: Use the angle sum property of a triangle
The sum of the angles in a triangle is always 180 degrees:
\[ x + y + z = 180 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 1 into Equation 3
Substituting \( z \) from Equation 1 into Equation 3:
\[ x + y + 2(x + y) = 180 \]
This simplifies to:
\[ x + y + 2x + 2y = 180 \]
\[ 3x + 3y = 180 \]
Dividing the entire equation by 3:
\[ x + y = 60 \quad \text{(Equation 4)} \]

### Step 5: Substitute Equation 4 into Equation 1
Now, substitute \( x + y = 60 \) back into Equation 1 to find \( z \):
\[ z = 2(x + y) = 2(60) = 120 \]

### Step 6: Find the smallest angle using Equation 2
Now, substitute \( z = 120 \) into Equation 2 to find \( x \):
\[ x = \frac{1}{4}z = \frac{1}{4}(120) = 30 \]

### Step 7: Find \( y \) using Equation 4
Now we can find \( y \) using Equation 4:
\[ x + y = 60 \implies 30 + y = 60 \implies y = 60 - 30 = 30 \]

### Conclusion: All angles in degrees
Thus, the angles of the triangle are:
- Smallest angle \( x = 30^\circ \)
- Second angle \( y = 30^\circ \)
- Largest angle \( z = 120^\circ \)

### Final Answer
The angles of the triangle are \( 30^\circ, 30^\circ, \) and \( 120^\circ \).

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