Question

which statements must be true about the reflection of
$\triangle xyz$ across $overleftrightarrow{mn}$? select three options.
$square m\angle xzy = 90^{circ}$
$square m\angle mcy = 90^{circ}$
$square \overline{xx} \cong \overline{yy}$
$square bz \cong bz$
$square xy \parallel xy$

Explanation:

Step1: Properties of reflection

When a figure is reflected across a line, the line of reflection is the perpendicular bisector of the segment joining a point and its image.

Step2: Analyze each option

• For \(m\angle XZY = 90^{\circ}\): There is no information from the reflection property to support this.
• For \(m\angle MCY=90^{\circ}\): Since \(MN\) is the line of reflection, and \(Y\) and \(Y'\) are reflection images, \(MN\perp YY'\), so \(m\angle MCY = 90^{\circ}\).
• For \(XX'\cong YY'\): The distance from a point to the line of reflection is the same as the distance from its image to the line of reflection. But \(XX'\) and \(YY'\) are not necessarily congruent.
• For \(BZ\cong BZ'\): Because \(Z\) and \(Z'\) are reflection images across \(MN\), and \(B\) is on \(MN\) (the line of reflection), by the property of reflection, \(BZ = BZ'\).
• For \(XY\parallel X'Y'\): When a figure is reflected, the pre - image and the image are congruent and corresponding sides are parallel.

Answer:

\(m\angle MCY = 90^{\circ}\), \(BZ\cong BZ'\), \(XY\parallel X'Y'\)