Question

triangle jkl and triangle jqr are drawn below. if \\(\overline{kl} \cong \overline{qr}\\), \\(\angle k \cong \angle q\\), and \\(\angle kjl \cong \angle qjr\\), use the dropdown boxes below to determine a sequence of two transformations that maps triangle jkl onto triangle jqr. then use a congruence statement to explain why this is possible.
translate triangle jkl until point aligns with point .
then rotate triangle jkl counterclockwise about point until has the same slope as .

Explanation:

Step1: Translation

Translate triangle \(JKL\) until point \(L\) aligns with point \(R\). This is because \(\overline{KL}\cong\overline{QR}\), so moving \(L\) to \(R\) (since \(KL\) and \(QR\) are congruent sides) starts to align the two triangles.

Step2: Rotation

Then rotate triangle \(J'K'L'\) (the translated triangle) counter - clockwise about point \(J\) until \(\overline{J'K'}\) has the same slope as \(\overline{JQ}\). Given \(\angle KJL\cong\angle QJR\) and \(\angle K\cong\angle Q\), after translation and rotation, the triangles will map onto each other.

Since \(\angle K\cong\angle Q\), \(\angle KJL\cong\angle QJR\), and \(\overline{KL}\cong\overline{QR}\), by the \(AAS\) (Angle - Angle - Side) congruence criterion, \(\triangle JKL\cong\triangle JQR\).

Answer:

Translate triangle \(JKL\) until point \(L\) aligns with point \(R\). Then rotate triangle \(J'K'L'\) counter - clockwise about point \(J\) until \(\overline{J'K'}\) has the same slope as \(\overline{JQ}\). The congruence statement is \(\triangle JKL\cong\triangle JQR\) (by \(AAS\)).