For this assignment, you should define what you think an angle is, and use your definition to prove the Vertical Angle Theorem.
In doing this, you can use David Henderson's questions as a guide,
but you need to integrate your answers to the various questions into a
unified, coherent whole. You should try to make your proof as clear
and convincing as possible. The best proofs are those that not only
convince the readers that something is true, but also allow them to understand
it is true. You should bear this in mind when you're deciding what
assumptions to make and what kind of a proof to give. This is one
reason that you want your assumptions to be simpler than the fact that
you're trying to prove, and as obvious as possible. (It's also one
reason that different people will give different proofs if they think of
angles in different ways. Explaining why two angles have the same
shape is very different than explaining why they have the same degree measure.)
If you do decide to give a proof using degree measure, be careful not to
confuse an angle (which is some kind of geometric object) with its degree
measure (which is a number). If you want to talk about measuring
angles, you'll need to explain how you want to assign a measurement to
each angle, and also why angle measurements under your definition have
any properties that that you need for your proof. This is one reason
that Henderson suggests that proofs using symmetries are"generally simpler"
than proofs involving measuring angles.