Question

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

Explanation:

Step1: Rotate triangle \(JKL\)

Rotate triangle \(JKL\) counter - clockwise about point \(L\) (not \(K\)) until \(\overline{KL}\) aligns with \(\overline{GL}\). Since \(\overline{KL}\cong\overline{GL}\), this rotation will make \(K\) coincide with \(G\) in terms of the side - length relationship.

Step2: Translate the rotated triangle

Then translate the rotated triangle \(J'K'L'\) (after rotation about \(L\)) so that the other corresponding parts match. Given \(\overline{JK}\cong\overline{FG}\) and \(\angle K\cong\angle G\), by the Side - Angle - Side (SAS) congruence criterion.

The congruence statement is \(\triangle JKL\cong\triangle FGL\) (by SAS: \(\overline{JK}\cong\overline{FG}\), \(\angle K\cong\angle G\), \(\overline{KL}\cong\overline{GL}\)) which means there is a sequence of rigid transformations (rotation and translation) that can map \(\triangle JKL\) onto \(\triangle FGL\)

Answer:

First, rotate triangle \(JKL\) counter - clockwise about point \(L\) until \(\overline{KL}\) aligns with \(\overline{GL}\). Then translate the rotated triangle. The congruence statement \(\triangle JKL\cong\triangle FGL\) (by SAS) shows the sequence of transformations (rotation and translation, which are rigid transformations) is possible as rigid transformations preserve congruence.