Question

question 6 (multiple choice worth 1 points) (03.03er.nc) shawn drew \\( \triangle lmn \\), in which \\( m\angle lmn = 90^\circ \\). he then drew \\( \triangle pqr \\), which was a dilation of \\( \triangle lmn \\) by a scale factor of 3 from the center of dilation at point m. which of these can be used to prove \\( \triangle lmn \sim \triangle pqr \\) by the aa similarity postulate? \\( \overline{lm} = 3\overline{pq} \\); this can be confirmed translating point p to point l \\( \overline{mn} = 3\overline{qr} \\); this can be confirmed translating point r to point n \\( m\angle p \cong m\angle n \\); this can be confirmed by translating point p to point n. \\( m\angle r \cong m\angle n \\); this can be confirmed by translating point r to point n.

Explanation:

Step1: Recall AA similarity postulate

AA (Angle - Angle) similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Step2: Analyze angle - angle congruence

Since \(\triangle PQR\) is a dilation of \(\triangle LMN\) from the center of dilation at point \(M\), \(\angle M\) is common to both triangles. Also, \(\angle LMN=\angle PQR = 90^{\circ}\) (right - angle).

Step3: Check side - angle - side (incorrect for AA)

The statements about side lengths (\(\overline{LM}=3\overline{PQ}\) and \(\overline{MN}=3\overline{QR}\)) are about side - side - side (SSS) similarity (if we consider the ratio of sides) or SAS similarity (if we consider the ratio of sides and included angle). But AA similarity only requires two angles.

Step4: Check angle - side (incorrect for AA)

The statements about angles \(m\angle R\cong m\angle N\) and \(m\angle P\cong m\angle N\) are not relevant for AA similarity as we already have two right angles and a common angle.

Answer:

The fact that \(\angle LMN=\angle PQR = 90^{\circ}\) (right - angle) and \(\angle M\) is common to both \(\triangle LMN\) and \(\triangle PQR\) (two angles congruent) can be used to prove \(\triangle LMN\sim\triangle PQR\) by the AA similarity postulate.