Question

which statements must be true about the reflection of △xyz across mn? select three options.
□m∠xzy = 90°
□m∠mcy = 90°
□xx ≅ yy
□bz ≅ bz
□xy || xy

Explanation:

Step1: Recall properties of reflection

A reflection across a line is a rigid - motion. Corresponding segments are congruent and the line of reflection is the perpendicular bisector of the segments joining corresponding points.

Step2: Analyze angle \(m\angle XZ'Y'\)

There is no information given to suggest that \(m\angle XZ'Y' = 90^{\circ}\) in general for a reflection.

Step3: Analyze angle \(m\angle MCY\)

There is no clear indication that \(m\angle MCY = 90^{\circ}\) based on the reflection properties.

Step4: Analyze \(\overline{XX'}\cong\overline{YY'}\)

In a reflection, the segments joining pre - image and image points of non - collinear points are not necessarily congruent. There is no reason for \(\overline{XX'}\cong\overline{YY'}\) to be true in general.

Step5: Analyze \(\overline{BZ'}\cong\overline{BZ}\)

Since the line of reflection \(\overleftrightarrow{MN}\) is the perpendicular bisector of the segment joining a point and its image, and \(B\) lies on the line of reflection \(\overleftrightarrow{MN}\), for point \(Z\) and its image \(Z'\), \(\overline{BZ'}\cong\overline{BZ}\) because \(B\) is on the line of reflection and the line of reflection is the perpendicular bisector of \(\overline{ZZ'}\).

Step6: Analyze \(\overline{XY}\parallel\overline{X'Y'}\)

In a reflection, corresponding sides of the pre - image and image triangles are parallel. So \(\overline{XY}\parallel\overline{X'Y'}\). Also, since the line of reflection \(\overleftrightarrow{MN}\) is the perpendicular bisector of the segments joining corresponding points, if we consider the fact that the transformation is a rigid motion, we can conclude that \(\overline{XY}\parallel\overline{X'Y'}\).

Answer:

\(\overline{BZ'}\cong\overline{BZ}\), \(\overline{XY}\parallel\overline{X'Y'}\) (assuming we need to pick two correct ones as the problem seems to have some incorrect initial guidance about picking three options based on the above analysis. If we must pick three, we need more information about the figure to make a third valid choice)