The law of sines states that:
[tex]\frac{A}{\sin\alpha}=\frac{B}{\sin\beta}=\frac{C}{\sin \gamma}[/tex]
Where alpha is the opposite angle to the A-side, beta is the opposite angle to the B-side, and gamma is the opposite angle to the C-side.
In our case
[tex]\begin{gathered} \frac{12}{\sin(80degree)}=\frac{8}{\sin \theta} \\ \Rightarrow\sin \theta=\frac{8\cdot\sin (80degree)}{12} \\ \Rightarrow\theta=\sin ^{-1}(\frac{8\cdot\sin(80degree)}{12}) \\ \Rightarrow\theta=\sin ^{-1}(\frac{2\cdot\sin (80degree)}{3}) \end{gathered}[/tex]
Then,
[tex]\begin{gathered} \theta=41.04\text{degre}e \\ \Rightarrow\angle K=41.04degree \end{gathered}[/tex]
Thus, angle K is 41.04°
As for angle M, remember that the sum of the inner angles of a triangle is equal to 180°, therefore:
[tex]\begin{gathered} \angle M=180-80-41.04=58.96 \\ \Rightarrow\angle M=58.96 \end{gathered}[/tex]
Then, angle M is equal to 58.96°.
Finally, as we know angle M, we can calculate the length of side m.
[tex]\begin{gathered} \frac{12}{\sin(80degree)}=\frac{m}{\sin (58.96degree)} \\ \Rightarrow m=\frac{12}{\sin(80degree)}\cdot\sin (58.96degree)=10.44 \\ \Rightarrow m=10.44 \end{gathered}[/tex]
Then, m is equal to 10.44
