Question

Which theorem or postulate proves the two triangles are similar

Answer 1

SSS - Theorem 

All the sides to you in the problem are given to you and no angles. 
The sides also proportioned out to each other by 1/2. 
In other words, the triangle on the right is 1/2 smaller than the one on the left 

Answer 2

Because all 3 sides are given for both triangles you can use the SSS theorem