Diagonals of Parallelogram Bisect Each Other

In  this post, we show that the diagonals of a parallelogram bisect each other. We will use the fact that the diagonal of a parallelogram is a transversal to the opposite sides of the parallelogram. The transversal and the sides form congruent alternate interior angles. Recall that the alternate interior angles formed by a transversal and parallel lines are congruent.

parallelogra-diagonals

Given

Parallelogram ABCD with diagonals \overline{AC} and \overline{BD}

What to Show

\overline{AO} \cong \overline{OC} and \overline{BO} \cong \overline{OD}.

Proof

\angle OAB \cong \angle OCD because they are alternate interior angles of parallel lines that pass through \overline {AB} and \overline {CD} cut by the  transversal \overline{AC} (A).

\overline{AB} \cong \overline{CD} because opposite sides of a parallelogram are congruent (S).

\angle ABO \cong \angle ODC because they are alternate interior angles of parallel lines that pass through \overline {AB} and \overline {CD} cut by the  transversal \overline{BD} (A).

By ASA Triangle Congruence theorem, \triangle AOB \cong \triangle COD.

\overline{AO} \cong \overline{OC} and \overline{BO} \cong \overline{OD} because corresponding sides of congruent triangles are congruent.

Related Posts Plugin for WordPress, Blogger...

Leave a Reply