Question

triangle congruence: asa and aas
what are the rigid transformations that will map △abc to △def?
translate vertex a to vertex d, and then rotate △abc around point a to align the sides and angles.
translate vertex a to vertex d, and then reflect △abc across the line containing ac
translate vertex b to vertex d, and then reflect △abc across the line containing ac
translate vertex b to vertex d, and then rotate △abc around point b to align the sides and angles.

Explanation:

Step1: Analyze triangle angles

In $\triangle ABC$, $\angle A = 40^\circ$, $\angle C = 67^\circ$, so $\angle B = 180^\circ - 40^\circ - 67^\circ = 73^\circ$. In $\triangle DEF$, $\angle D = 40^\circ$, $\angle E = 73^\circ$, so $\triangle ABC \cong \triangle DEF$ (ASA or AAS).

Step2: Check transformation steps

First, translate vertex $A$ to $D$ (aligning the $40^\circ$ angles). Then, reflect $\triangle ABC$ across the line containing $AC$? No, wait—wait, the correct transformation: Translate vertex $A$ to $D$, then rotate $\triangle ABC$ around point $A$ (now $D$) to align sides/angles? Wait, no, let's re - evaluate. Wait, the correct option: Translate vertex $A$ to $D$, and then reflect $\triangle ABC$ across the line containing $AC$? No, wait the option: "Translate vertex A to vertex D, and then rotate $\triangle ABC$ around point A to align the sides and angles"—wait, no, let's check the angle correspondence. $\angle A=\angle D = 40^\circ$, $\angle B=\angle E = 73^\circ$, $\angle C=\angle F = 67^\circ$. So first, translate $A$ to $D$. Then, since $\angle B$ and $\angle E$ are equal, we need to rotate around $A$ (now $D$) to align the sides. Wait, the correct option is the one with "Translate vertex A to vertex D, and then rotate $\triangle ABC$ around point A to align the sides and angles" (the top - right option). Wait, let's check the options again. The options are:

1. Translate vertex A to vertex D, and then rotate $\triangle ABC$ around point A to align the sides and angles.

1. Translate vertex B to vertex D, and then reflect $\triangle ABC$ across the line containing $AC$.

1. Translate vertex B to vertex D, and then rotate $\triangle ABC$ around point B to align the sides and angles.

1. Translate vertex A to vertex D, and then reflect $\triangle ABC$ across the line containing $AC$.

Wait, first, $\angle A=\angle D = 40^\circ$, side $AB$ and $DE$—wait, $AB$ has a mark, $DE$ has a mark. So $AB = DE$. So translate $A$ to $D$, then rotate around $A$ (now $D$) to align the other sides/angles. So the correct transformation is: Translate vertex $A$ to vertex $D$, and then rotate $\triangle ABC$ around point $A$ to align the sides and angles.

Answer:

The correct option is the one with the description "Translate vertex A to vertex D, and then rotate $\triangle ABC$ around point A to align the sides and angles" (the top - right card in the image).