Answer:
The measure of angle C is;
[tex]\measuredangle C=35.7^{\circ}[/tex]Explanation:
Given the figure in the attached image.
To get angle C, we will need to first calculate the length of side a.
Applying cosine rule;
[tex]a=\sqrt[]{b^2+c^2-2bc\cos A}[/tex]given;
[tex]\begin{gathered} b=12 \\ c=7 \\ A=55^{\circ} \end{gathered}[/tex]substituting the given values;
[tex]\begin{gathered} a=\sqrt[]{12^2+7^2-2(12)(7)\cos 55} \\ a=9.83 \end{gathered}[/tex]We can now solve for angle C;
[tex]\begin{gathered} \cos C=\frac{a^2+b^2-c^2}{2ab} \\ \text{substituting;} \\ \cos C=\frac{9.83^2+12^2-7^2}{2(9.83)(12)} \\ \cos C=0.81226 \\ C=\cos ^{-1}(0.81226) \\ C=35.7^{\circ} \end{gathered}[/tex]Therefore, the measure of angle C is;
[tex]\measuredangle C=35.7^{\circ}[/tex]