Determine the series of transformations that map polygon ABCDE onto polygon A'B'C'D'E'.

To determine a series of transformations that would map polygon ABCDE onto polygon A'B'C'D'E', you would follow these steps:

1. Identify the orientation: Check if polygon ABCDE has the same orientation as polygon A'B'C'D'E' (oriented the same way, meaning not flipped), or if it is a mirror image (reflected).

2. Compare sizes: Look at the two polygons and determine if they are the same size or if one is larger or smaller than the other.

3. Determine the type of transformations needed based on the observations:

- If the polygons are the same size and orientation, you may only need a translation (sliding the polygon) or a rotation to map one onto the other. - If the polygons are the same size but with a different orientation (one being the mirror image of the other), a reflection over a line is needed. - If the polygons are different sizes but have the same orientation, a dilation (resizing) is needed along with potentially translation and rotation. - If the polygons are different sizes and different orientations, you may need a series of transformations, including reflection, translation, rotation, and dilation.

4. Apply the transformations in sequence: a. If a reflection is needed, reflect ABCDE over the appropriate line to match the orientation of A'B'C'D'E'. b. If a dilation is necessary, scale ABCDE up or down from a center point (could be a vertex or a point external to the polygon) until it is the same size as A'B'C'D'E'. c. Translate the polygon by moving it along a vector from one vertex to the corresponding vertex of the other polygon (e.g., from A to A’). d. If a rotation is needed, rotate the polygon around a point (usually one of its vertices or the center of dilation) by the angle that aligns it with A'B'C'D'E'.

By carefully analyzing the orientation, size, and position of both polygons, you can determine the exact sequence and types of transformations required to map polygon ABCDE onto polygon A'B'C'D'E'.