Parallelograms: Practice

A parallelogram is defined by both pairs of opposite sides being parallel. Turning the page changes none of those relationships.

Side $AD$ is opposite $BC$, so those lengths match.

Angles $A$ and $C$ are opposite. Opposite angles of a parallelogram are congruent.

Angles $D$ and $A$ are consecutive, so they are supplementary.

The intersection bisects diagonal $AC$, so its two parts are congruent.

Point $E$ is the midpoint of $BD$.

The point $O$ is the midpoint of $AC$. A half-turn sends $A$ along line $AC$ to the equally distant point on the other side of $O$, which is $C$.

Bisection compares the two parts of each diagonal separately. It does not compare the full diagonals. A general oblique parallelogram can have unequal diagonal lengths.

For example, a parallelogram can have these diagonal lengths:

A segment has its own length in either triangle. That common included side, together with two alternate interior-angle pairs, gives ASA.

Bisection means each diagonal is cut into two equal parts. For diagonal $BD$, those parts are $BE$ and $ED$.

Because the parallelogram is oblique, its two angle sizes differ. A symmetry preserves the diagonal intersection, so a nonidentity rotational symmetry has that intersection as its center. It must send a corner to the equal opposite corner, since the neighboring angles differ and it cannot fix a corner. This forces the half-turn. The two rotation amounts, counting the identity, are

Our definitions are inclusive. A square has both pairs of opposite sides parallel, so it is a parallelogram and inherits every direct property proved here.