Yes, they do exist. In fact, there are infinitely many of them.
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The five shapes that have one line of symmetry are equilateral triangle, isosceles triangle, scalene triangle, rectangle, and rhombus. A shape has one line of symmetry if it can be folded along a line so that the two halves match exactly. In the case of these shapes, there is only one line that can divide the shape into two congruent halves.
Well, darling, shapes with only one line of symmetry are like the lone rangers of the symmetry world. We're talking about your squares, rectangles, and isosceles triangles strutting their stuff with just one line to keep them in check. So, if you're looking for a shape that's a bit of a rebel but still plays by the rules, these are the ones for you.
They have a bilateral symmetry that means that there is only one way you can cut them to get 2 equal shapes.
Right triangles can have only one line of symmetry. Only right triangles with two 45 degree angles can have a line of symmetry.
It depends on what kind of shape you are asking about. A circle has an infinite number of lines of symmetry, all passing through the center. Regular shapes (triangle, square, rectangle, pentagon, etc.) have at least one. Irregular shapes tend to have no line of symmetry.