Construct and angle and bisect it, that should do it.
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A vertex.
Two angles are adjacent if they have a common side and a common vertex.
Yes. Consider the two angles in the letter Z (or N). The diagonal line is common but the angles do not share a vertex.
They are adjacent angles.
No, complementary angles do not need to have the same vertex. Complementary angles are comprised of any two angles whose sum is 90 degrees. The definition of a complementary angle does not say that it needs to have the same vertex.