The circular measure of an angle of an isosceles triangle is `(5pi)/(9)` . Circular measure of one of the other angles must be

Options

A. `(5pi)/(18)`
B. `(5pi)/(9)`
C. `(2pi)/(9)`
D. `(4pi)/(9)`

Answer

To solve the problem, we need to find the circular measure of one of the equal angles in an isosceles triangle, given that one angle is \( \frac{5\pi}{9} \).

### Step-by-Step Solution:

1. **Identify the Angles in the Isosceles Triangle**:
Let the triangle be \( ABC \) where \( AB = AC \). This means that angles \( B \) and \( C \) are equal. We can denote these equal angles as \( X \). The given angle \( A \) is \( \frac{5\pi}{9} \).

2. **Use the Angle Sum Property**:
The sum of the angles in any triangle is \( \pi \) radians (which is equivalent to 180 degrees). Therefore, we can write:
\[ A + B + C = \pi \]
Substituting the known values:
\[ \frac{5\pi}{9} + X + X = \pi \]
This simplifies to:
\[ \frac{5\pi}{9} + 2X = \pi \]

3. **Rearranging the Equation**:
To isolate \( 2X \), we subtract \( \frac{5\pi}{9} \) from both sides:
\[ 2X = \pi - \frac{5\pi}{9} \]

4. **Finding a Common Denominator**:
To perform the subtraction, we need a common denominator. The value of \( \pi \) can be expressed as \( \frac{9\pi}{9} \):
\[ 2X = \frac{9\pi}{9} - \frac{5\pi}{9} \]
This simplifies to:
\[ 2X = \frac{4\pi}{9} \]

5. **Solving for \( X \)**:
Now, divide both sides by 2 to find \( X \):
\[ X = \frac{4\pi}{9} \div 2 = \frac{4\pi}{18} = \frac{2\pi}{9} \]

6. **Conclusion**:
The circular measure of one of the equal angles \( B \) or \( C \) in the isosceles triangle is:
\[ \frac{2\pi}{9} \]

### Final Answer:
The circular measure of one of the other angles must be \( \frac{2\pi}{9} \).