Question

describe a composition of rigid motions that maps de to jk and ∠d to ∠j. why does this composition show that there is no angle - side - side congruence criterion? translate point d onto point then △def about point such that point e lies on point even though ef≅kl, point f will not map to point to prove the triangles congruent.

Explanation:

Step1: Translate point D

Translate $\triangle DEF$ so that point D maps onto point J. This is the first - step in the rigid - motion composition.

Step2: Rotate about point J

Rotate $\triangle DEF$ (after translation) about point J so that $\overline{DE}$ maps onto $\overline{JK}$. Since $\angle D\cong\angle J$, this rotation is possible.

Step3: Check congruence

After the translation and rotation, we have two pairs of corresponding sides equal ($\overline{DE}\cong\overline{JK}$ and $\overline{EF}\cong\overline{KL}$) and the included angles equal ($\angle D\cong\angle J$). By the Side - Angle - Side (SAS) congruence criterion, the triangles are congruent.

Answer:

First, translate point D onto point J. Then, rotate $\triangle DEF$ about point J so that $\overline{DE}$ maps onto $\overline{JK}$. The triangles $\triangle DEF$ and $\triangle JKL$ are congruent by the Side - Angle - Side (SAS) congruence criterion because after the rigid - motion composition, we have two pairs of corresponding sides equal and the included angles equal.