Let’s consider each statement one by one to determine which ones must be true regarding triangles [tex]\( \triangle XYZ \)[/tex] and [tex]\( \triangle X'Y'Z' \)[/tex] after a reflection over a vertical line followed by a dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex].
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
Reflecting a triangle over a vertical line does not change the shape of the triangle, only its orientation. Dilation changes the size of the triangle but maintains the shape and proportions. Hence, the two triangles will be similar.
Conclusion: [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex] is true.
2. [tex]\( \angle X Z Y = \angle Y Z X \)[/tex]
This statement says that two specific angles within the same triangle are equal. Without detailed information about the angles in the given triangles, we must assume generic angle properties in geometry.
In any triangle, the following is generally not true unless specifically stated:
- [tex]\( \angle X Z Y \neq \angle Y Z X \)[/tex] unless the triangle has a particular symmetry or equal distribution, which is rare.
Conclusion: [tex]\( \angle X Z Y = \angle Y Z X \)[/tex] is not true generally.
3. [tex]\( \overline{Y X} \cong \overline{Y X} \)[/tex]
Congruence of the line segment [tex]\( \overline{YX} \)[/tex] before and after dilation would typically imply that their lengths remain the same. However, dilation by [tex]\( \frac{1}{2} \)[/tex] changes the length of the line segment by scaling it down to half its original length. Therefore, congruence cannot hold true.
Conclusion: [tex]\( \overline{Y X} \cong \overline{Y X} \)[/tex] is false.
4. [tex]\( xz = 2 \times 2z \)[/tex]
This statement seems to contain some unclear notation. If we interpret the statement [tex]\( xz \)[/tex] and [tex]\( 2z \)[/tex] as coordinates or values directly scaled by dilation, the notation itself seems incorrect. Additionally, this does not comply with standard geometric transformations.
Conclusion: [tex]\( xz = 2 \times 2z \)[/tex] is false.
5. [tex]\( m \angle Y X Z = 2 \, m \angle Y X X \)[/tex]
Dilation affects the lengths of the sides of a triangle but not the measurements of the angles. Hence, the measure of an angle before and after dilation remains the same. This means that the angle measure would not be doubled.
Conclusion: [tex]\( m \angle Y X Z = 2 \, m \angle Y X X \)[/tex] is false.
Based on these evaluations, the correct choices are:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
2. [tex]\( \overline{Y X} \cong \overline{Y X} \)[/tex] (False)
3. [tex]\( xz = 2 \times 2z \)[/tex] (False)
Since we need three true options and have identified only one, the only valid true option is:
- [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
Hence, the complete correct set is:
(True, True, False, False, False) for these conditions.
