Using the Law of Sines to solve the all possible triangles if $\angle A=114^{\circ}, a=26, b=12$. If no answer exists, enter DNE for all answers. degrees; degrees; Assume $\angle A$ is opposite side $a, \angle B$ is opposite side $b$, and $\angle C$ is opposite side $c$.

Step 1 ：First, we use the Law of Sines to find \(\angle B\). The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. This can be written as: \(\frac{a}{\sin A} = \frac{b}{\sin B}\)

Step 2 ：We can rearrange this to solve for \(\sin B\): \(\sin B = \frac{b \sin A}{a}\)

Step 3 ：Substituting the given values: \(\sin B = \frac{12 \sin 114^\circ}{26} \approx 0.434\)

Step 4 ：Since the sine function only returns values between -1 and 1, and our calculated value is within this range, we can find the angle B using the inverse sine function: \(B = \sin^{-1}(0.434) \approx 26^\circ\)

Step 5 ：However, since the sine function is positive in both the first and second quadrants, there is a second possible value for angle B in the second quadrant: \(B' = 180^\circ - 26^\circ = 154^\circ\)

Step 6 ：Now, we can find \(\angle C\) by subtracting \(\angle A\) and \(\angle B\) from 180°: \(C = 180^\circ - A - B = 180^\circ - 114^\circ - 26^\circ = 40^\circ\)

Step 7 ：and \(C' = 180^\circ - A - B' = 180^\circ - 114^\circ - 154^\circ = -88^\circ\)

Step 8 ：Since an angle cannot be negative in a triangle, there is no triangle with \(\angle B = 154^\circ\)

Step 9 ：So, the only possible triangle has \(\angle A = 114^\circ\), \(\angle B = 26^\circ\), \(\angle C = 40^\circ\)

Step 10 ：The final answer is \(\boxed{\angle A = 114^\circ, \angle B = 26^\circ, \angle C = 40^\circ}\)