Two different triangles have side lengths of 13 and 16 units and a non included angle of 50 explain how the triangles are different

  see the attachments

Step-by-step explanation:

Actually, the given description fits 3 different triangles. They differ in the size of the included angle, hence the length of the remaining side. Possibilities are ...

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When the non-included angle is opposite the long side, there is only one solution to the triangle. (See the first attachment.)

When the non-included angle is opposite the shorter of the given sides, there are generally two possible solutions. One of these has an acute angle opposite the long side; the other has an obtuse angle opposite the long side.

In any event, the Law of Sines is used to solve the triangle. First, the angle opposite the other given side is found; then the remaining angle and the side opposite it.

Suppose the given sides are "a" and "b", and the given angle is A. Then the Law of Sines tells you ...

  B = arcsin(sin(A)·b/a) . . . . . possibly 2 values

From which ...

  C = 180° -A -B

which allows finding ...

  c = a·sin(C)/sin(A)