It is the singular of the plural vertices relating to angles of polygons
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Adjacent angles are defined as angles that share a side and a vertex point. A vertex is a corner point.
Adjacent angles have a shared vertex.
A vertex.
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.
Yes all inscribed angles in a circle have their vertex on the circumference of the circle. Central angles have their vertex at the center of the circle.