To determine which transformation could be used to prove the figures are similar using the [tex]$AA$[/tex] (Angle-Angle) similarity postulate, let's understand the effect of dilation and the properties required for [tex]$AA$[/tex] similarity.
First, dilation is a transformation that alters the size of a figure but preserves the shape, specifically the angle measures within the figure. When a figure is dilated by a scale factor of [tex]\(\frac{1}{3}\)[/tex], each side length of the figure is reduced to one-third, but all angles remain the same.
The [tex]$AA$[/tex] similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Similar triangles have the same shape but not necessarily the same size, meaning their corresponding angles are equal, and their corresponding sides are proportional.
Given that dilation preserves angle measures, we need a transformation that can utilize the preserved angles to establish similarity through [tex]$AA$[/tex] similarity postulate.
Among the given choices:
1. A translation can map corresponding angles of one figure onto another because a translation preserves both angle measures and lengths of sides. However, side lengths are not needed directly for proving similarity using [tex]$AA$[/tex] similarity postulate, just the angles.
2. A rotation can change the orientation of a figure but also preserves angle measures and lengths of sides. Since orientation is not crucial for similarity through [tex]$AA$[/tex] similarity, rotation is not the necessary transformation.
3. A dilation directly affects side lengths and angle measures, but since it says it can map one side onto another, it's not focusing on angle preservation, which is essential here.
4. A reflection changes the orientation of a figure, just like rotation, but it does not address the preservation of angle measures specifically required for proving similarity through [tex]$AA$[/tex] similarity.
Since dilations preserve the angle measures of a figure, the most appropriate transformation to utilize these preserved angle measures for [tex]$AA$[/tex] similarity postulate is:
A translation because it can map one angle onto another since dilations preserve angle measures of triangles.
So, we choose the first option:
* A translation because it can map one angle onto another since dilations preserve angle measures of triangles.
Therefore, the correct answer is 1.
