Question

given: $\overline{bd}$ and $\overline{ac}$ bisect each other.
prove: $\overline{ab}\parallel\overline{cd}$.
note: quadrilateral properties are not permitted in this proof
step
statement
reason
1
$\overline{bd}$ and $\overline{ac}$ bisect each other
given

Explanation:

Step1: Identify congruent triangles

Since \(\overline{BD}\) and \(\overline{AC}\) bisect each other, we have \(AE = EC\) and \(BE=ED\). Also, \(\angle AEB=\angle CED\) (vertical angles are equal). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle AEB\cong\triangle CED\).

Step2: Use CPCTC

By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), \(\angle EAB=\angle ECD\).

Step3: Apply the converse of the alternate interior angles theorem

If two lines are cut by a transversal such that alternate interior angles are equal, then the lines are parallel. Here, \(\overline{AB}\) and \(\overline{CD}\) are cut by transversal \(\overline{AC}\), and \(\angle EAB=\angle ECD\) (alternate interior angles). So, \(\overline{AB}\parallel\overline{CD}\).

Answer:

\(\overline{AB}\parallel\overline{CD}\) is proved.