Trigonometry/Geometry - Law of sines and cosines

In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles ABC and DEF such that AB = DE, BC = EF, and angle A = angle D, then we cannot deduce that ABC and DEF are congruent.

However, there are a few special cases in which SSA "works". That is, suppose we have AB = DE = x, BC = EF=y, and \angle A = \angle D = \theta. For some values of x, y, and \theta, we can deduce that triangle ABC is congruent to triangle DEF. Use the Law of Cosines or Law of Sines to explain the conditions x, y, and/or theta must satisfy in order for us to be able to deduce that triangle ABC is congruent to triangle DEF. (In other words, find conditions on x, y, and theta, so that given these values, you can uniquely reconstruct triangle ABC.)

Thanks to bobpursley I found the answer to the LAw of cosines, that if theta = 90 then both triangles are equal. But what about the law of sines, and any other ones? Thanks, Help is appreciated

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asked by Sam
  1. Determine if there are zero, one or two possible triangles. Drae the triangles, if possible, including the unknown measurements.
    The question is, In triangle JKL, angle J is 55, side j is 10.4 and side k is 11.6.

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