An isosceles triangle has one line of symmetry, which is drawn from the noncongruent side to the opposite vertex, and does not have a rotation symmetry.
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You cannot have one parallel line. There must be another line that it is parallel to. Also, 1 symmetry is meaningless - you can have one axis of symmetry or rotational symmetry of order one. The answer will depend on which one it is.
Rotational symmetry of order 1.
NO
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Every right circular cone, conic section, and regular polygon has at least one line of symmetry.