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Both laws are same and equivalent.  If you have a triangle, then we can use the triangle law.  If you have a parallelogram, use that law.

In general the parallelogram law is used, because it is easier to understand and apply.

If you have a parallelogram  ABCD,  then vector AB + vector BC = vector AC. AC is the third side of triangle ABC.  This is triangle law.

vector AD + vector AB = vector AC.   
this is parallelogram law.

(but we have vector AD = vector BC..)

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