The symmetric property of equality says that if a = b, then b = a.
The associative property for addition merely states that a+(b+c) = (a+b)+c.
The distributive property says that if a, b, and c are real numbers, then
a · (b + c) = (a · b) + (a · c).
The transitive property of equality says that if a = b and b = c, then a = c.
Consider statament: If x + y = z and z = x + 4, then x + y = x + 4. Denote a=x+y, b=z and c=x+4. Now you have if a=b and b=c, then a=c. This is exactly transitive property.