Question

prove the triangles are congruent, given \\( \overline { a b } \cong \overline { c d } \\) and \\( m \\) is the midpoint of \\( \overline { a d } \\& \overline { b c } \\)

Explanation:

Step1: Mid - point property

Since \(M\) is the mid - point of \(BC\), by the definition of a mid - point, \(BM = CM\). So \(BM\cong CM\).

Step2: Mid - point property

Since \(M\) is the mid - point of \(AD\), by the definition of a mid - point, \(AM=DM\). So \(AM\cong DM\).

Step3: Vertical angles

\(\angle AMB\) and \(\angle DMC\) are vertical angles. By the vertical angles theorem, \(\angle AMB\cong\angle DMC\).

Step4: Congruence criterion

We have \(AB\cong CD\) (given), \(BM\cong CM\), \(AM\cong DM\). Using the Side - Side - Side (SSS) congruence criterion, \(\triangle ABM\cong\triangle DCM\).

Answer:

\(BM\cong CM\) (because \(M\) is the mid - point of \(BC\)), \(AM\cong DM\) (because \(M\) is the mid - point of \(AD\)), \(\triangle ABM\cong\triangle DCM\) (by SSS congruence).