Explanation
Using usual abbreviations for sectors
$$A_1 = A_2$$ [Given]
$$\Rightarrow \dfrac{\pi r^2_1 \theta_1}{360^\circ} = \dfrac{\pi r^2_2 \theta_2}{360^\circ}$$
$$\Rightarrow r^2_1 \theta_1 = r^2_2\theta_2$$
$$\Rightarrow \dfrac{\theta_1}{\theta_2} = \dfrac{r^2_2}{r^2_1}$$
Now, $$\dfrac{l_1}{l_2} = \dfrac{2\pi r_1\theta_1}{2\pi r_2\theta_2} = \dfrac{r_1}{r_2} \times \dfrac{r^2_2}{r^2_1} = \dfrac{r_2}{r_1}$$
$$\Rightarrow \dfrac{l_1}{l_2} = \dfrac{r_2}{r_1}$$
Hence, arcs length can be equal if $$\dfrac{r_2}{r_1} = 1$$ i.e., $$r_1 = r_2 = r.$$
Hence, the given statement is false.
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