Question

what are the rigid transformations that will map △abc to △def? translate vertex a to vertex d, and then reflect △abc across the line containing $overline{ac}$. translate vertex b to vertex d, and then rotate △abc around point b to align the sides and angles. translate vertex b to vertex d, and then reflect △abc across the line containing $overline{ac}$. translate vertex a to vertex d, and then rotate △abc around point a to align the sides and angles.

Explanation:

1. First, analyze the angles and sides of the triangles. In $\triangle ABC$, $\angle A = 40^\circ$, $\angle C=67^\circ$, so $\angle B = 180 - 40 - 67=73^\circ$. In $\triangle DEF$, $\angle D = 40^\circ$, $\angle E = 73^\circ$, so the triangles are congruent.
2. To map $\triangle ABC$ to $\triangle DEF$, we can start by translating vertex $A$ to vertex $D$ (since $\angle A$ and $\angle D$ are both $40^\circ$). Then, rotating $\triangle ABC$ around point $A$ will align the sides and angles because after translation, the corresponding angles and sides can be aligned by rotation.
3. Let's check other options:

• Option 1: Reflecting across $\overline{AC}$ after translating $A$ to $D$ won't align the triangles correctly as the angle and side relationships don't match the reflection.
• Option 2: Translating $B$ to $D$ and rotating around $B$ is incorrect because $\angle B$ and $\angle D$ are not the same in terms of the mapping ( $\angle B = 73^\circ$, $\angle D = 40^\circ$).
• Option 3: Translating $B$ to $D$ and reflecting across $\overline{AC}$ is incorrect for similar reasons as the angle and vertex correspondences are wrong.
• Option 4: Translating $A$ to $D$ (matching the $40^\circ$ angle) and then rotating around $A$ to align the sides and angles (since the other angles and sides will match after rotation) is correct.

Answer:

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