How to determine the range of a angle measure?

Question

In $\Delta$ $KLM$, $KL=20$ $LM=13$ m$\angle K$$=40$. What is the range for angle $M$'s measure? Something like between $90^{\circ}$ and $180^{\circ}$.

Answer 1

You can use the law of sine, then find the value exact of angle measure

$$\frac{\sin{(M)}}{20}=\frac{\sin{(K)}}{13}$$

Answer 2

Drop the altitude $LS$ of the isoceles triangle $\triangle M_1LM_2$.

Then $LS=13\sin\phi$. On the other hand, since $\sin K=\frac{LS}{KL}$ we have $\sin 40^{\circ}=\frac{13\sin\phi}{20}$. Hence, $\sin\phi=\frac{20\sin40^{\circ}}{13}\approx 0.9889$ and $\phi\approx 81.4568^{\circ}$ (the acute angle solution).

The obtuse angle solution is $180^{\circ}-\phi\approx 98.5432^{\circ}.$