To find the measure of the tangent-tangent angle, we can follow these steps:
1. Understand the property of the tangent-tangent angle:
The measure of the tangent-tangent angle is equal to half the difference of the measures of the intercepted arcs.
2. Identify the intercepted arcs:
Given arcs measure [tex]\(149^{\circ}\)[/tex] and [tex]\(211^{\circ}\)[/tex].
3. Find the difference between the intercepted arcs:
Calculate the absolute difference between the two arcs:
[tex]\[
|211^{\circ} - 149^{\circ}| = 62^{\circ}
\][/tex]
4. Calculate half of this difference:
The measure of the tangent-tangent angle is half the difference of the intercepted arcs:
[tex]\[
\frac{1}{2} \times 62^{\circ} = 31^{\circ}
\][/tex]
Thus, the measure of the tangent-tangent angle is [tex]\(31^{\circ}\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{31^{\circ}}
\][/tex]
Therefore, the answer is option C.
