If the length of an arc of a circle of radius $r$ is equal to that of an arc of a circle of radius $2r$,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?

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(A) The statement is true,not false.
Let two circles $C_1$ and $C_2$ have radii $r$ and $2r$ with centers $O$ and $O'$,respectively.
It is given that the arc length $l_1$ of $C_1$ is equal to the arc length $l_2$ of $C_2$,i.e.,$l_1 = l_2 = l$.
Let $\theta_1$ be the angle subtended by the arc of $C_1$ and $\theta_2$ be the angle subtended by the arc of $C_2$ at their respective centers.
The formula for arc length is $l = \frac{\theta}{360^\circ} \times 2\pi R$.
For $C_1$: $l = \frac{\theta_1}{360^\circ} \times 2\pi r$ ........... $(i)$
For $C_2$: $l = \frac{\theta_2}{360^\circ} \times 2\pi(2r) = \frac{\theta_2}{360^\circ} \times 4\pi r$ ........... $(ii)$
Equating $(i)$ and $(ii)$:
$\frac{\theta_1}{360^\circ} \times 2\pi r = \frac{\theta_2}{360^\circ} \times 4\pi r$
$\theta_1 = 2\theta_2$
Thus,the angle of the sector of the first circle is double the angle of the sector of the second circle. Therefore,the statement is true.

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