Question

which statements must be true about the image of δmnp after a reflection across ↔eg? select three options. the image will be congruent to δmnp. the orientation of the image will be the same as the orientation of δmnp. ↔eg will be perpendicular to the line segments connecting the corresponding vertices. the line segments connecting the corresponding vertices will all be congruent to each other. the line segments connecting corresponding vertices will all be parallel to each other.

Explanation:

1. Congruence in Reflection: A reflection is a rigid transformation, so the image of $\triangle MNP$ after reflection across $\overleftrightarrow{EG}$ will be congruent to $\triangle MNP$. This is a fundamental property of reflections (rigid transformations preserve size and shape, hence congruence).
2. Orientation in Reflection: Reflections reverse the orientation of the figure (e.g., a clockwise - oriented figure becomes counter - clockwise after reflection). So the orientation of the image will not be the same as that of $\triangle MNP$.
3. Perpendicularity of Axis and Segments: The line of reflection ($\overleftrightarrow{EG}$) is the perpendicular bisector of the segments joining corresponding vertices. So $\overleftrightarrow{EG}$ is perpendicular to the line segments connecting corresponding vertices.
4. Congruence of Corresponding Segments: The segments connecting corresponding vertices are all bisected by the line of reflection and are congruent to each other because the distance from each vertex to the line of reflection is equal to the distance from its image to the line of reflection, so the length of each segment (between a vertex and its image) is twice the distance from the vertex to the line of reflection, and thus all these segments are congruent.
5. Parallelism of Corresponding Segments: The segments connecting corresponding vertices are all perpendicular to the line of reflection (from the property of reflection), so they are parallel to each other (since they are all perpendicular to the same line).

Answer:

• The image will be congruent to $\triangle MNP$.
• $\overleftrightarrow{EG}$ will be perpendicular to the line segments connecting the corresponding vertices.
• The line segments connecting the corresponding vertices will all be parallel to each other.
• The line segments connecting the corresponding vertices will all be congruent to each other.

(Note: There are actually four correct statements here, but if we follow the "select three" instruction, we can choose any three of the four correct ones. However, based on the properties: congruence of image, perpendicularity of axis to connecting segments, and parallelism (or congruence) of connecting segments. The most accurate set of three would be: "The image will be congruent to $\triangle MNP$", "$\overleftrightarrow{EG}$ will be perpendicular to the line segments connecting the corresponding vertices", and "The line segments connecting the corresponding vertices will all be parallel to each other" (or "The line segments connecting the corresponding vertices will all be congruent to each other").)