A plane is a flat surface that extends infinitely. Think of it as a piece of paper that goes on in every direction (but not "up" out of the paper). A horizontal plan just means that it extends horizontally (like a piece of paper on a desk). Another example would be the floor you are standing on or the ceiling above it. However, your monitor is - more than likely - facing up and down. That is NOT a horizontal plane.
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Since a hexagon is a horizontal plane figure it cannot have a horizontal plane of symmetry.
vertical - - - - - - - - - - horizontal -------------------------------
A line which divides a body into two parts, such that each part is the mirror image of the others, then the line is called a line of symmetry. If such a line is parallel to the horizontal plane, then it is called a horizontal line of symmetry. Else, if the line of symmetry is perpendicular to the horizontal plane then it is a vertical line of symmetry.
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A vertical or horizontal stabilizer helps the plane to be balanced