Question

which statements must be true about the reflection of δxyz across ↔mn? select three options. □ ( mangle xzy = 90^circ ) □ ( mangle mcy = 90^circ ) □ ( overline{xx} cong overline{yy} ) □ ( overline{bz} cong overline{bz} ) □ ( overline{xy} parallel overline{xy} )

Explanation:

1. For \( m\angle X'Z'Y' = 90^\circ \): Reflection preserves angle measures, so if \( \angle XZY = 90^\circ \) (implied by the diagram's right - angled triangle), then \( m\angle X'Z'Y'=m\angle XZY = 90^\circ \).
2. For \( \overline{BZ'}\cong\overline{BZ} \): In a reflection across a line (here \( \overleftrightarrow{MN} \)), the line of reflection is the perpendicular bisector of the segment joining a point and its image. So, \( B \) lies on \( \overleftrightarrow{MN} \), and \( Z' \) is the image of \( Z \) under reflection across \( \overleftrightarrow{MN} \), which means \( BZ' = BZ \), so \( \overline{BZ'}\cong\overline{BZ} \).
3. For \( \overline{XY}\parallel\overline{X'Y'} \): Reflection across a line preserves the orientation and parallelism in the sense that the image of a segment is parallel to the original segment when the line of reflection is not intersecting the segment (or due to the properties of isometry in reflection). Also, since \( XX' \) and \( YY' \) are both perpendicular to \( \overleftrightarrow{MN} \), \( XX'\parallel YY' \), and by the properties of reflection, \( XY\parallel X'Y' \).

• \( m\angle MCY = 90^\circ \): There is no reason to assume that \( \angle MCY \) is a right angle from the reflection properties.
• \( \overline{XX'}\cong\overline{YY'} \): \( XX' \) and \( YY' \) are both perpendicular to \( \overleftrightarrow{MN} \), but their lengths depend on the distance of \( X \) and \( Y \) from \( \overleftrightarrow{MN} \). There is no guarantee that they are congruent.

Answer:

• \( m\angle X'Z'Y' = 90^\circ \)
• \( \overline{BZ'}\cong\overline{BZ} \)
• \( \overline{XY}\parallel\overline{X'Y'} \)