Derek wrote the following paragraph proof for the vertical angles theorem:
line segment nt intersects line segment mr forming four angles. angles 1 and 3 are vertical angles. angles 2 and 4 are vertical angles.
the sum of angle 1 and angle 4 and the sum of angle 3 and angle 4 are each equal to 180 degrees by the definition of supplementary angles. the sum of angle 1 and angle 4 is equal to the sum of angle 3 and angle 4 by the transitive property of equality. angle 1 is equal to angle 3
which phrase completes the proof?