Is the choice vertical angle theorem correct?

Unfortunately it is incorrect. You should instead go for the Alternate Exterior Angle Theorem

Angle 2 and angle 7 are on opposite or alternate sides of the transversal cut. They are also exterior because they are on the outside of the train tracks (think of line a and line b as the metal rail of the train tracks). If a || b, then the alternate exterior angles are congruent, which applies to angle 2 and angle 7. Also it applies to angle 1 and angle 8. 

Best answer that i found on yahoo ok   set :   x= length of 2nd piece in cm  x- 3= length of 1st piece in cm  x-4= length of 3rd piece in cm  then since you know they all came from the 20 cm ribbon you add them together and set them equal to 20x+ x-3 +x-4 = 20  3x - 7= 20  3x= 27  x= 9  the shortest piece is the third piece which is x- 4  x-4  9-4 = 5  the length of the smallest piece is 5 cm.  the other lengths are  9 cm  6 cm