Definition: A parallelogram is a quadrilateral with two pairs of parallel sides; its opposite sides are congruent, its opposite angles are congruent, and its diagonals bisect each other.
What properties does it have?
- Opposite sides are parallel and equal.
- Opposite angles are equal; consecutive angles add to 180 degrees.
- The diagonals bisect each other at their midpoints.
How do you prove a quadrilateral is a parallelogram?
Show any one of these: both pairs of opposite sides equal, both pairs of opposite angles equal, diagonals bisecting each other, or one pair of sides both equal and parallel.
Example. Quadrilateral ABCD has A(1, 1), B(6, 2), C(8, 6), D(3, 5). The midpoint of AC is (4.5, 3.5). The midpoint of BD is (4.5, 3.5). The diagonals share a midpoint, so ABCD is a parallelogram. Also, AB has slope 1/5 and DC has slope (6 – 5)/(8 – 3) = 1/5, confirming a parallel pair.
What goes wrong?
Assuming diagonals are equal or perpendicular, which holds only for special cases such as rectangles and rhombuses. Another error is that one pair of equal sides alone is not enough, because an isosceles trapezoid has that too.