Yes, the vertices of all cones are single points and, since points are geometric entities with 0 dimensions, they are all congruent.
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The horizontal cross-sections of a cone are circular in shape, and these circles are congruent to each other at all heights except for the vertex, which is a single point. As you move away from the vertex along the height of the cone, the diameter of the circular cross-sections increases uniformly. This consistent shape results in a series of congruent circles, illustrating the cone's geometric properties.
No, a cone has one vertex.
A cone only has 1 vertex because it only has one point where it connects. (vertex=a point)
You could call the vertex, the apex, the tip, or the point of the cone.
No, there is no reason for a cone and a cylinder to have anything congruent.