Let θ be a positive angle. If the number of degrees in θ is divided...

From going by the options, option (A), $\theta=30^{{}^{\circ}}$, as we know that $180^{{}^{\circ}}=\pi$ radian $\therefore \quad 30^{{}^{\circ}}=\frac{30 \pi}{180}$ radian Now according to question, $\frac{30^{{}^{\circ}} \times 180^{{}^{\circ}}}{30^{{}^{\circ}} \pi}=\frac{180}{\pi}$ Now number of degree in $\theta$ is multiplied by number of radians in $\theta$. $\therefore \quad 30^{{}^{\circ}} \times \frac{30 \pi}{180}=\frac{900 \pi}{180}=\frac{10 \pi}{2}=5 \pi \neq \frac{125 \pi}{9}$ From option (B), $$\begin{aligned} & \theta=45^{{}^{\circ}} \\ \therefore \quad & 45^{{}^{\circ}}=\frac{45^{{}^{\circ}} \pi}{180} \text{ radian }\end{aligned}$$ Now according to question, $\frac{45^{{}^{\circ}} \times 180}{45^{{}^{\circ}} \pi}=\frac{180}{\pi}$ Now number of degree in $\theta$ is multiplied by number of radian in $\theta$. $\therefore \quad 45^{{}^{\circ}} \times \frac{45^{{}^{\circ}} \pi}{180^{{}^{\circ}}}=\frac{45 \pi}{4} \neq \frac{125 \pi}{9}$ From option (C), $\theta=50^{{}^{\circ}}$ As we know that $180^{{}^{\circ}}=\pi$ radian $\therefore \quad 50^{{}^{\circ}}=\frac{50 \pi}{180}$ radian Now according to question $\frac{50^{{}^{\circ}} \times 180^{{}^{\circ}}}{50^{{}^{\circ}} \pi}=\frac{180}{\pi}$ Now number of degree in ' $\theta$ ' is multiplied by number of radian in $\theta$. $\therefore \quad 50^{{}^{\circ}} \times \frac{50 \pi}{180}=\frac{2500 \pi}{180}=\frac{125 \pi}{9}\therefore \quad$ Option (C) is correct.