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: Use mid - point property for \(BC\)

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

Step2: Use mid - point property for \(AD\)

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

Step3: Use SSS (Side - Side - Side) congruence criterion

In \(\triangle ABM\) and \(\triangle DCM\), we have \(AB\cong CD\) (given), \(BM\cong CM\) (from mid - point of \(BC\)), \(AM\cong DM\) (from mid - point of \(AD\)). By SSS congruence criterion, \(\triangle ABM\cong\triangle DCM\)

Answer:

\(BM\cong CM\) (Reason: Definition of mid - point for \(BC\)); \(AM\cong DM\) (Reason: Definition of mid - point for \(AD\)); \(\triangle ABM\cong\triangle DCM\) (Reason: SSS (Side - Side - Side) congruence criterion)