Proofs about parallelograms
A parallelogram is defined only by its parallel sides. How do its equal sides and halved diagonals follow, and how do you prove a shape is one?
Where you are 4 inside
What this area is about
The definition of a parallelogram says only that both pairs of opposite sides are parallel. Every other fact about it has to be proved, and nearly every proof uses one move: draw a diagonal, find two congruent triangles, and read off their matching parts.
How these fit together
Properties of every parallelogram. Proving properties of a parallelogram: opposite sides are congruent, by ASA across one diagonal, and the diagonals bisect each other, using both diagonals and the first theorem.
Running them backwards. Proving a quadrilateral is a parallelogram turns those properties into tests: both pairs of opposite sides congruent, both pairs of opposite angles congruent, diagonals bisecting each other, or one pair of opposite sides both parallel and congruent, \overline{AB} \parallel \overline{DC} and \overline{AB} \cong \overline{DC}.
Special parallelograms. Proving a parallelogram is a rectangle or a rhombus adds one fact: a right angle or congruent diagonals for a rectangle; two congruent neighboring sides, perpendicular diagonals or a diagonal that bisects an angle for a rhombus; one of each for a square.
A near relative. Proving properties of an isosceles trapezoid: only one pair of sides is parallel; the same method plus two altitudes shows that congruent legs force congruent base angles and diagonals.
A path through it
Properties, tests, then the rectangle and the rhombus. The trapezoid goes last: it uses the parallelogram tests and, for its converses, the rectangle.
The same facts also yield to coordinate proofs, where midpoints, lengths and slopes replace the triangles.