Question

given $overline{ac}$ and $overline{bd}$ bisect each other
prove: $overline{ad}paralleloverline{bc}$
which of the following statements is not needed, if the proof makes use of the other three?
$ab = dc$
$\triangle aedcong\triangle ceb$
$mangle aeb = mangle dec$
$ae = ec$ and $de = eb$

Explanation:

Step1: Use vertical angles

Since \(\angle AED=\angle BEC\) (vertical angles are equal).

Step2: Use given bisect condition

Given \(AE = EC\) and \(DE=EB\), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle AED\cong\triangle CEB\).

Step3: Use congruent triangle properties

If \(\triangle AED\cong\triangle CEB\), then \(\angle ADE=\angle CBE\) (corresponding parts of congruent triangles).

Step4: Use alternate interior angles

Since \(\angle ADE\) and \(\angle CBE\) are alternate interior angles, \(AD\parallel BC\) (if alternate interior angles are equal, then the lines are parallel).

Now, for the statement \(AB = DC\):
If we use the proof based on \(\triangle AED\cong\triangle CEB\) (using \(AE = EC\), \(DE = EB\), \(\angle AED=\angle BEC\)), we do not need the information \(AB = DC\) to prove \(AD\parallel BC\).

Answer:

\(AB = DC\)