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 is a rigid - motion. Rigid motions preserve angle measures, lengths of segments, and parallelism.

Step2: Analyze angle - measure preservation

Since reflection preserves angle measures, if in $\triangle XYZ$, $\angle XZY$ has a certain measure, in its reflection $\triangle X'Z'Y'$, $m\angle X'Z'Y'$ will be the same as $m\angle XZY$. But we have no information that $\angle XZY = 90^{\circ}$, so we can't say $m\angle X'Z'Y'=90^{\circ}$. Also, there is no information to suggest $m\angle MCY = 90^{\circ}$.

Step3: Analyze segment - length preservation

In a reflection, the distance from a point to the line of reflection is the same as the distance from its image to the line of reflection. Points $B$ is on the line of reflection $MN$. For any point $Z$ and its image $Z'$, the segment from $Z$ to $B$ and from $Z'$ to $B$ are equal in length, so $BZ = BZ'$.

Step4: Analyze segment - length for non - on - reflection - line points

For points $X$ and $Y$ and their images $X'$ and $Y'$, the segments $XX'$ and $YY'$ are perpendicular to the line of reflection $MN$. But there is no reason for $XX'=YY'$ in general.

Step5: Analyze parallelism

Since reflection preserves parallelism, if two lines are parallel in the pre - image, they are parallel in the image. $\overline{XY}$ and $\overline{X'Y'}$ are corresponding sides of the pre - image and image of a reflection, so $\overline{XY}\parallel\overline{X'Y'}$.

Answer:

$BZ = BZ'$, $\overline{XY}\parallel\overline{X'Y'}$