Question

triangle opq and triangle jkl are drawn below. if \\(\overline{op} \cong \overline{jk}\\), \\(\overline{pq} \cong \overline{kl}\\), and \\(\overline{oq} \cong \overline{jl}\\), use the dropdown boxes below to determine a sequence of two transformations that maps triangle opq onto triangle jkl. then use a congruence statement to explain why this is possible.

Explanation:

Step1: Rotate triangle \(OPQ\)

Rotate triangle \(OPQ\) so that the orientation of the sides matches the general orientation of triangle \(JKL\) better. For example, rotate it \(180^{\circ}\) (the specific rotation degree can be adjusted according to the relative position of the two triangles in the coordinate - like concept).

Step2: Translate the rotated triangle

Translate (slide) the rotated triangle \(OPQ\) (now \(O'P'Q'\)) so that the corresponding congruent sides (\(\overline{OP}\cong\overline{JK}\), \(\overline{PQ}\cong\overline{KL}\), \(\overline{OQ}\cong\overline{JL}\)) overlap.

Since \(\overline{OP}\cong\overline{JK}\), \(\overline{PQ}\cong\overline{KL}\), and \(\overline{OQ}\cong\overline{JL}\), by the Side - Side - Side (SSS) congruence criterion. If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So, \(\triangle OPQ\cong\triangle JKL\). A congruence transformation (a combination of rotation and translation, which are rigid motions) can map one congruent triangle onto another.

Answer:

First, rotate triangle \(OPQ\) (by an appropriate angle, say \(180^{\circ}\) depending on the visual orientation). Then, translate the rotated triangle. Since \(\triangle OPQ\cong\triangle JKL\) (by SSS: \(\overline{OP}\cong\overline{JK}\), \(\overline{PQ}\cong\overline{KL}\), \(\overline{OQ}\cong\overline{JL}\)), a sequence of rotation and translation (rigid motions) can map \(\triangle OPQ\) onto \(\triangle JKL\).