Question

given ( overline{lm} ), ( overline{mn} ), ( overline{nl} ) are midsegments, which similarity statement is true? ( \triangle rqpsim\triangle lnm ) ( \triangle rqpsim\triangle mlq ) ( \triangle rqpsim\triangle rnm ) ( \triangle rqpsim\triangle npl )

Explanation:

Step1: Recall the mid - segment theorem

The mid - segment of a triangle is parallel to the third side and half its length. Since \(LM\), \(MN\), \(NL\) are mid - segments, \(LM\parallel PR\), \(MN\parallel PQ\), \(NL\parallel QR\).

Step2: Use the AA (Angle - Angle) similarity criterion

For \(\triangle RQP\) and \(\triangle LNM\):

• \(\angle Q=\angle Q\) (common angle).
• Because \(LM\parallel PR\) and \(NL\parallel QR\), \(\angle QLM=\angle QPR\) and \(\angle QNL=\angle QRP\) (corresponding angles).

By AA similarity criterion, \(\triangle RQP\sim\triangle LNM\).

For \(\triangle RQP\) and \(\triangle MLQ\):

• \(\angle Q\) is common. But the other angles are not equal as the side - angle relationships do not match for similarity.

For \(\triangle RQP\) and \(\triangle RNM\):

• \(\angle R\) is common. But the side - angle relationships (due to mid - segments) do not satisfy the similarity criteria.

For \(\triangle RQP\) and \(\triangle NPL\):

• \(\angle P\) is not common with \(\triangle RQP\) in a way that the AA criterion is met.

Answer:

\(\triangle RQP\sim\triangle LNM\)