Question

determining the number of possible triangles
in △mno, m = 20, n = 14, and m∠m = 51°. how many distinct triangles can be formed given these measurements?
there are no triangles possible.
there is only one distinct triangle possible, with m∠n ≈ 33°.
there is only one distinct triangle possible, with m∠n ≈ 147°.
there are two distinct triangles possible, with m∠n ≈ 33° or m∠n ≈ 147°.

Explanation:

Step1: Apply the Law of Sines

By the Law of Sines, $\frac{\sin N}{n}=\frac{\sin M}{m}$. Substitute $m = 20$, $n = 14$, and $M=51^{\circ}$. So $\sin N=\frac{n\sin M}{m}=\frac{14\sin51^{\circ}}{20}$.

Step2: Calculate $\sin N$ value

$\sin N=\frac{14\times0.777}{20}=\frac{10.878}{20}=0.5439$.

Step3: Find angle $N$

$N=\sin^{- 1}(0.5439)\approx33^{\circ}$. Also, since $\sin\theta=\sin(180^{\circ}-\theta)$, another possible value of $N$ is $180 - 33=147^{\circ}$. But when $M = 51^{\circ}$ and $N=147^{\circ}$, $M + N=51^{\circ}+147^{\circ}=198^{\circ}>180^{\circ}$, which is not possible in a triangle. So only the $N = 33^{\circ}$ case is valid, and there is only one distinct triangle possible with $m\angle N\approx33^{\circ}$.

Answer:

There is only one distinct triangle possible, with $m\angle N\approx33^{\circ}$.