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

Answer

To find the third angle of the triangle given one angle in degrees and another angle in radians, we will follow these steps:

### Step 1: Convert the angle from radians to degrees.
We have one angle given as \(\frac{\pi}{4}\) radians. To convert this to degrees, we use the conversion formula:

\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]

Substituting \(\frac{\pi}{4}\) into the formula:

\[ \text{Degrees} = \frac{\pi}{4} \times \frac{180}{\pi} = \frac{180}{4} = 45^\circ \]

### Step 2: Identify the angles in the triangle.
Now we have:
- Angle A = \(54^\circ\)
- Angle B = \(45^\circ\)

### Step 3: Use the triangle angle sum property.
The sum of the angles in a triangle is always \(180^\circ\). Therefore, we can set up the equation:

\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]

Substituting the known angles:

\[ 54^\circ + 45^\circ + \text{Angle C} = 180^\circ \]

### Step 4: Solve for the third angle.
Now, we can simplify the equation:

\[ 99^\circ + \text{Angle C} = 180^\circ \]

Subtract \(99^\circ\) from both sides:

\[ \text{Angle C} = 180^\circ - 99^\circ = 81^\circ \]

### Step 5: Convert the third angle to centesimal units.
To convert degrees to centesimal units, we use the conversion factor where \(1^\circ = \frac{10}{9}\) grades (centesimal units):

\[ \text{Angle C in centesimal} = 81^\circ \times \frac{10}{9} = 90 \text{ grades} \]

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

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