In a tringle the measure of the greatest angle is square of measure of the smallest angle and the other angle is double of the smaller angle. Find the greatest angle of the triangle.

Answer

To solve the problem step by step, we will denote the angles of the triangle as follows:

Let:
- \( C \) = the smallest angle
- \( B \) = the other angle, which is double the smallest angle, so \( B = 2C \)
- \( A \) = the greatest angle, which is the square of the smallest angle, so \( A = C^2 \)

### Step 1: Set up the equation using the triangle angle sum property
The sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:

\[ A + B + C = 180 \]

Substituting the expressions for \( A \) and \( B \):

\[ C^2 + 2C + C = 180 \]

This simplifies to:

\[ C^2 + 3C = 180 \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:

\[ C^2 + 3C - 180 = 0 \]

### Step 3: Factor the quadratic equation
Next, we need to factor the quadratic equation \( C^2 + 3C - 180 = 0 \). We look for two numbers that multiply to -180 and add up to 3. The numbers are 15 and -12. Thus, we can factor the equation as:

\[ (C + 15)(C - 12) = 0 \]

### Step 4: Solve for \( C \)
Setting each factor to zero gives us:

1. \( C + 15 = 0 \) → \( C = -15 \) (not valid since angles cannot be negative)
2. \( C - 12 = 0 \) → \( C = 12 \)

### Step 5: Find the other angles
Now that we have \( C = 12 \) degrees, we can find \( B \) and \( A \):

- \( B = 2C = 2 \times 12 = 24 \) degrees
- \( A = C^2 = 12^2 = 144 \) degrees

### Step 6: Identify the greatest angle
The greatest angle \( A \) is:

\[ A = 144 \text{ degrees} \]

### Final Answer
The greatest angle of the triangle is **144 degrees**.

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