Are two opposite rays with common endpoint considered to have an angle? Why?

Yes.

An angle is formed by any two lines(or rays) meeting at a  common point. It has a measure.They can be added . They can be subtracted.

Proof:

Let AO and BO be opposite rays meeting at O.

Construction: Let a ray from CO come from C and meet at O.

Now AOB is a straifght line as  two opposite rays meet at O.

But AOC is an angle, AO and CO are two rays meeting at O

BOC is another angle., for the similar reason.

But AOC+BOC = AOC. Therefore, AOC being the sum of two angles is also an angle.

Therefore, opposite rays meeting at a common point is considered to be  anangle , or rather is an angle.

When the two  rays AO  and BO  meet at O, there is an angle contained beween them AO and BO (or OA and OB)

Keeping BO fixed, O of OA fixed  on O of OB ,you can move OA about O like a compass. For any position A, there is an angle between the arms AOB. Therefore, When AO comes exacctly opposite to BO,  AOB  becomes one sraight light line. Even then  there is measure of angle berween AO and BO and it is a special angle called straight angle measuring 180degrees (or Pi radians).