Question

determining the number of possible triangles
how many distinct triangles can be formed for which m∠e = 64°, g = 9, and e = 10?
triangle(s)
how many distinct triangles can be formed for which m∠j = 129°, k = 8, and j = 3?
triangle(s)

Explanation:

Step1: Use the Law of Sines

The Law of Sines is $\frac{\sin E}{e}=\frac{\sin G}{g}$. For the first - case with $E = 64^{\circ},g = 9,e = 10$. We first find $\sin G$: $\sin G=\frac{g\sin E}{e}$.

Step2: Calculate $\sin G$

Substitute the values: $\sin G=\frac{9\sin64^{\circ}}{10}$. Since $\sin64^{\circ}\approx0.8988$, then $\sin G=\frac{9\times0.8988}{10}=0.80892$. Since $0\lt\sin G\lt1$, and $e\gt g$, there are two possible triangles.

Step3: For the second - case

Using the Law of Sines $\frac{\sin J}{j}=\frac{\sin K}{k}$ with $J = 129^{\circ},k = 8,j = 3$. We find $\sin K=\frac{k\sin J}{j}$.

Step4: Calculate $\sin K$

Substitute the values: $\sin K=\frac{8\sin129^{\circ}}{3}$. Since $\sin129^{\circ}\approx0.7771$, then $\sin K=\frac{8\times0.7771}{3}\approx2.0723\gt1$. Since the range of the sine function is $[- 1,1]$, no triangle can be formed.

Answer:

2
0