Once you have proved that the reflection image of a segment...

Once you have proved that the reflection image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof. $$\text { If } \overline{A^{\prime} B^{\prime}} \text { is the reflection of } \overline{A B}, \text { then } A B=A^{\prime} B^{\prime}$$

Once you have proved that the reflection image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof.
$$\text { If } \overline{A^{\prime} B^{\prime}} \text { is the reflection of } \overline{A B}, \text { then } A B=A^{\prime} B^{\prime}$$

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Key Concepts

Reflection Transformation

A reflection is a type of isometry in Euclidean geometry that maps every point of a figure to another point such that the line of reflection is the perpendicular bisector of the segment joining corresponding points. This property ensures that distances between points and angles between segments are preserved, thereby producing a mirror image of the original figure.

Congruence of Geometric Figures

In Euclidean geometry, congruence means that two figures are identical in shape and size, though they may differ in position or orientation. When dealing with segments, congruence implies that the segment and its reflected image have equal lengths, which is a direct consequence of the distance preserving property of the reflection transformation.

Properties of Isometries

Isometries, which include reflections, rotations, and translations, are transformations that preserve distances and angle measures. Understanding that these transformations maintain the geometric properties of figures provides the foundation for proving that the reflection of a segment has the same length as the original segment, supporting the conclusion that the segments are congruent.

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