Question

which statements are true about reflections? check all that apply.
an image created by a reflection will always be congruent to its pre-image.
an image and its pre-image are always the same distance from the line of reflection.
if a point on the pre-image lies on the line of reflection, the image of that point is the same as the pre-image.
the line of reflection is perpendicular to the line segments connecting corresponding vertices.
the line segments connecting corresponding vertices are all congruent to each other.
the line segments connecting corresponding vertices are all parallel to each other.

Explanation:

1. Congruence in Reflection: A reflection is a rigid transformation, so the image and pre - image are always congruent. So the statement "An image created by a reflection will always be congruent to its pre - image" is true.
2. Distance from Line of Reflection: By the definition of a reflection, for any point in the pre - image, its image is the mirror - image across the line of reflection. This means that the distance from a point in the pre - image to the line of reflection is equal to the distance from its image to the line of reflection. So "An image and its pre - image are always the same distance from the line of reflection" is true.
3. Point on Line of Reflection: If a point lies on the line of reflection, when we reflect it, it maps onto itself. So "If a point on the pre - image lies on the line of reflection, the image of that point is the same as the pre - image" is true.
4. Perpendicularity of Line of Reflection: The line of reflection is the perpendicular bisector of the segment joining a point and its image. So the line of reflection is perpendicular to the line segments connecting corresponding vertices. This statement is true.
5. Congruence of Connecting Segments: The line segments connecting corresponding vertices are all congruent because each of these segments is bisected by the line of reflection and the length from a point to the line of reflection is equal to the length from its image to the line of reflection. So the length of each segment (from pre - image point to image point) is \(2\times\) (distance from pre - image point to line of reflection), and since corresponding points are equidistant from the line of reflection, these segments are congruent. So this statement is true.
6. Parallelism of Connecting Segments: The line segments connecting corresponding vertices are not parallel. They are perpendicular to the line of reflection (or bisected by it at right angles). So this statement is false.

Answer:

• An image created by a reflection will always be congruent to its pre - image.
• An image and its pre - image are always the same distance from the line of reflection.
• If a point on the pre - image lies on the line of reflection, the image of that point is the same as the pre - image.
• The line of reflection is perpendicular to the line segments connecting corresponding vertices.
• The line segments connecting corresponding vertices are all congruent to each other.