None. A line of symmetry, if it exists, is infinitely long and so a third of the line cannot exist as a separate entity.
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A square has 4 lines of symmetry
You would call the group of shapes which has at least one line of symmetry symmetrical shapes. Stars, circles, and rectangles are considered symmetrical shapes.
It all depends what types of shapes, but take for example a circle, it can have multiple lines of symmetry.
There are infinitely many such shapes. To start with, any regular will do.
Trapezoid, isosceles triangle, arrow, kite, etc.