If the length of an arc of a circle of radius $r$ is equal to that of an arc of a circle of radius $2 r$, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false? Why?

False

Let two circles $C _{1}$ and $C _{2}$ of radius $r$ and $2 r$ with centres $O$ and $O ^{\prime},$ respectively.

It is given that, the arc length $\widehat{A B}$ of $C_{1}$ is equal to arc length $\widehat{C D}$ of $C_{2}$ i.e.. $\widehat{A B}=\widehat{C D}=l$ (say)

Now, let $\theta_{1}$ be the angle subtended by arc $\widehat{A B}$ of $\theta_{2}$ be the angle subtended by arc $\widehat{C D}$ at the centre.

$\therefore \widehat{A B}=l=\frac{Q_{1}}{360} \times 2 \pi r$ ...........$(i)$

and $\widehat{C D}=l=\frac{\theta_{2}}{360} \times 2 \pi(2 r)=\frac{\theta_{2}}{360} \times 4 \pi r$ ...........$(ii)$

From Eqs. $(i)$ and $(ii),$

$\frac{\theta_{1}}{360} \times 2 \pi r=\frac{\theta_{2}}{360} \times 4 \pi r$

$\Rightarrow \quad \theta_{1}=2 \theta_{2}$

i.e., angle of the corresponding sector of $C _{1}$ is double the angle of the corresponding sector of $C _{2}$

It is true statement