One angle of a triangle is `54^(@)` and another angle `(pi)/(4)` radian. Find the third angle in centesimal unit.

Answer

To find the third angle of the triangle in centesimal units, we can follow these steps:

### Step 1: Identify the given angles
We have:
- Angle A = 54 degrees
- Angle B = \( \frac{\pi}{4} \) radians

### Step 2: Convert Angle B from radians to degrees
To convert radians to degrees, we use the conversion factor:
\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
Thus,
\[ \text{Angle B} = \frac{\pi}{4} \text{ radians} = \frac{\pi}{4} \times \frac{180}{\pi} = 45 \text{ degrees} \]

### Step 3: Use the triangle angle sum property
The sum of the angles in a triangle is always 180 degrees. Therefore, we can find the third angle (Angle C) using the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180 \text{ degrees} \]
Substituting the known values:
\[ 54 + 45 + \text{Angle C} = 180 \]

### Step 4: Solve for Angle C
Combine the known angles:
\[ 99 + \text{Angle C} = 180 \]
Now, isolate Angle C:
\[ \text{Angle C} = 180 - 99 = 81 \text{ degrees} \]

### Step 5: Convert Angle C from degrees to centesimal units
To convert degrees to centesimal units (gradians), we use the conversion:
\[ 90 \text{ degrees} = 100 \text{ gradians} \]
Thus, we can set up the proportion:
\[ \text{Angle C in gradians} = 81 \text{ degrees} \times \frac{100 \text{ gradians}}{90 \text{ degrees}} = \frac{81 \times 100}{90} \]
Calculating this gives:
\[ \text{Angle C in gradians} = \frac{8100}{90} = 90 \text{ gradians} \]

### Final Answer
The third angle in centesimal units is \( 90 \) gradians.

---