There are two circles such that the measure of the arc length formed by the first circle is equal to the measure of the arc length of the other circle. If the radius of the first circle is half of the other, then the relationship between the two angles is ${\mathrm{\theta }}_1=2{\mathrm{\theta }}_2$. 

Consider two circles of radii r and 2r respectively.

Let the length of two arcs be $l_1$ and $l_2$.

The length of an arc is given by the formula: 

$\frac{\theta }{360°}\times 2\pi r$

$\therefore l_1=\frac{2\pi r{\theta }_1}{360°}$ and,

$l_2$ $=\frac{2\pi r{\theta }_2}{360°}$

$=\frac{2\pi ·2r{\theta }_2}{360°}$

According to the statement given:

$l_1=l_2$

$\Rightarrow \frac{2\pi r{\theta }_1}{360°}=\frac{2\pi ·2r{\theta }_2}{360°}$

$\Rightarrow {\theta }_1=2{\theta }_2$

i.e., the angle of the sector of the first circle is twice the angle of the sector of the other circle.

Hence, the given statement is true.