list type of angles, with proves method of opposite angles theory

Describe (with an illustrative example/figure) three different meanings of angle (dynamic movement, measure, geometric shape) and the three corresponding meanings of congruent, one for each meaning of angle.
a) Prove the vertical angle theorem. Begin your proof with a clear statement of the theorem, which must include what is given and what is to be proven.
Prove the vertical angle theorem in a different way, using a different meaning of angle. Repeat all parts of question 2.
Prove the vertical angle theorem in a different way, using yet another meaning of angle. Repeat all parts of question 2.
b)Does your proof work on the surface of the sphere? Why or why not?
c)Which of the three meanings of angle from problem 1 are you primarily using in your proof? Briefly explain how this meaning is used in your proof.
Note: you must write three proofs one for each meaning of the angle (dynamic movement, measure, geometric shape)do not use outside sourses