Is the following a reflection or rotation

Question

To show that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent, we can use a series of transformations and apply the ASA (Angle-Side-Angle) Congruence Theorem. First, we need to identify the corresponding parts of the two triangles: \( \angle A \) matches with \( \angle D \), \( \angle B \) matches with \( \angle E \), and the side \( \overline{AB} \) corresponds to \( \overline{DE} \). To make the triangles align, we start with a translation, moving \( \triangle ABC \) so that point \( A \) is on point \( D \) and point \( B \) is on point \( E \). Next, we’ll rotate \( \triangle ABC \) around point \( D \) to line up \( \angle A \) with \( \angle D \). After this rotation, point \( C \) will fall into the position of point \( F \), making all the angles and sides match. Since translations and rotations do not change the shape or size of triangles, we can be sure that \( \triangle ABC \) is still congruent to \( \triangle DEF \). Finally, we confirm that we have two angles (\( \angle A \) with \( \angle D \) and \( \angle B \) with \( \angle E \)) and the side \( \overline{AB} \) with \( \overline{DE} \) that are all congruent, which fits the ASA theorem perfectly. In conclusion, by translating and rotating the triangles, we’ve shown that \( \triangle ABC \) is indeed congruent to \( \triangle DEF \).

Answer 1

The description you've provided outlines a series of transformations used to demonstrate the congruence of two triangles, \( \triangle ABC \) and \( \triangle DEF \). In this case, you specifically mention a translation followed by a rotation.

1. Translation: This is when you slide the triangle \( \triangle ABC \) so that point \( A \) is on point \( D \) and point \( B \) is on point \( E \). This is not a reflection or rotation; it simply moves the triangle without changing its orientation.

2. Rotation: After the translation, you rotate \( \triangle ABC \) around point \( D \) to align \( \angle A \) with \( \angle D \). A rotation involves turning the triangle around a fixed point (in this case, point \( D \)).

So, to directly answer your question: the transformation involves both a translation and a rotation. There is no mention of a reflection in your description. Both translations and rotations maintain the congruence of the triangles, ensuring that their shapes and sizes remain unchanged.