An Ellipse
It will be a hyperbola.
hyperbola
parabola
A transversal is simply any line that passes through two or more coplanar lines each at different points. So picture, if you will, two lines that are clearly not parallel. I can easily construct a transversal that passes through them. HOWEVER, if two parallel lines are intersected by a transversal, then the corresponding angles are congruent. This is called the transversal postulate. If the corresponding angles are congruent, than the lines are parallel. This is the converse of the first postulate. So, the answer to your question is NO, unless the corresponding angles are congruent.
An ellipse is produced.
A line is produced
An Ellipse
If I understand your description correctly, a line.
When a right circular cone is intersected by a plane that passes through its vertex and touches the edge of each nappe, the resulting shape is a triangle. This triangle is formed by the intersection line extending from the vertex to the edges of the cone's surfaces, effectively creating a triangular cross-section of the cone.
It will be a hyperbola.
hyperbola
hyperbola
If a right circular cone intersects a plane that runs parallel to the cone's axis but does not pass through its vertex, the resulting curve is a pair of hyperboles.
CIRCULAR
When it is produced it is processed through a tube, giving it a circular shape.
Providing that the lines are parallel that the transversal passes through then it will have two equal alternate angles that are on opposite sides of the transversal.