A parallelogram has no line of symmetry.
Statement: A parallelogram has no line of symmetry.
( extbf{Goal:} ) Understand why a usual parallelogram does not have any mirror line.
A line of symmetry (mirror line) splits a shape into two identical halves. If you fold the shape on that line, both halves match exactly.
( ext{Mirror line} Rightarrow ext{left half} = ext{right half (after fold)} )
(AB parallel CD)
(BC parallel AD)
Opposite sides are parallel and equal. Adjacent sides can be of different lengths and angles are usually slanted (not right angles).
(a) Try a line through midpoints of opposite sides (vertical or horizontal)
( ext{Line through midpoints of } AD ext{ and } BC Rightarrow ext{reflect } A leftrightarrow D ?)
( ext{But angles are slanted, so shapes don't overlap exactly.} )
Because the sides are slanted, folding on this line won’t make the two halves coincide.
(b) Try a diagonal as mirror line
( ext{Consider diagonal } AC )
( riangle ABC otcong riangle CDA ext{ by reflection} )
Across a diagonal, the angles and side positions don’t match like a mirror; they match by rotation, not reflection.
(c) Try the other diagonal
( ext{Consider diagonal } BD )
( riangle ABD otcong riangle CBD ext{ by reflection} )
Same issue: reflection fails; only a rotation of (180^circ) maps the shape to itself.
( ext{Parallelogram has } 180^circ ext{ rotational symmetry} )
( ext{but no reflection symmetry (no mirror line).} )
Only special parallelograms like rectangles, rhombuses, and squares have mirror lines.
( ext{General parallelogram} Rightarrow ext{no line of symmetry} )
( ext{Rectangle/Rhombus/Square} Rightarrow ext{have mirror lines} )
For a general parallelogram, no line can split it into two identical mirror halves. So the statement is true.