Given that the angles of the two sectors are equal, we can find the relationship between the angles, radii, and the lengths of the arc
The length of the arc (S) is given by the formula
[tex]S\text{ = }\frac{\theta}{360}\text{ x 2}\pi\text{ r}[/tex][tex]\text{Since 2}\pi=360^0[/tex][tex]S\text{ =}\theta\text{ r}[/tex]Then we can make the angle the subject of the formula
[tex]\theta\text{ =}\frac{S}{r}[/tex]For the first sector
[tex]\begin{gathered} \text{with radius r}_1\text{ and angle }\theta_1 \\ \\ \theta_1\text{ =}\frac{S_1}{r_1} \end{gathered}[/tex]For the second sector
[tex]\begin{gathered} \text{with radius r}_2\text{ and }\theta_2 \\ \theta_2=\frac{S_2}{r_2} \end{gathered}[/tex][tex]\theta_{1\text{ =}}\text{ }\theta_2\text{ = }\theta\text{ =}\frac{S_1}{r_1}\text{ =}\frac{S_2}{r_2}[/tex]Simplifying the equation, we will obtain
[tex]\theta\text{ =}\frac{S_1}{r_1}\text{ =}\frac{S_2}{r_2}[/tex]