To determine which statements must be true for the triangles [tex]\(\triangle XYZ\)[/tex] and [tex]\(\Delta X'Y'Z'\)[/tex], let's analyze the transformations step-by-step:
1. Reflection Over a Vertical Line:
- Reflections over vertical lines do not change the shape or size of the triangle. They only alter the orientation. Therefore, the corresponding sides and angles remain congruent after the reflection.
2. Dilation by a Scale Factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex] reduces all side lengths of the triangle by half.
- Dilation does not alter the measures of the angles within the triangle.
Given these transformations, let's review each of the provided statements:
1. [tex]\(\triangle XYZ \cong \triangle X'Y'Z'\)[/tex]:
- False. Dilation changes the size of the triangle, so [tex]\(\triangle XYZ\)[/tex] is not congruent to [tex]\(\triangle X'Y'Z'\)[/tex]. The angles remain the same, but the side lengths are different due to the scale factor.
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]:
- True. Reflections and dilations do not affect the measures of the triangle's angles. Therefore, the corresponding angles are congruent.
3. [tex]\(\overline{YX} \cong \overline{Y'X'}\)[/tex]:
- False. After dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex], the segments will be half the length of the original. Thus, [tex]\(\overline{YX}\)[/tex] is not congruent to [tex]\(\overline{Y'X'}\)[/tex].
4. [tex]\(XZ = 2X'Z'\)[/tex]:
- True. Since the triangle was dilated by a factor of [tex]\(\frac{1}{2}\)[/tex], the new side lengths are half the original lengths. Thus, [tex]\(XZ\)[/tex] should be twice as long as [tex]\(X'Z'\)[/tex].
5. [tex]\(m\angle YXZ = 2m\angle Y'X'Z'\)[/tex]:
- False. The measure of the angles is unchanged by either reflection or dilation. Therefore, the measures of [tex]\(\angle YXZ\)[/tex] and [tex]\(\angle Y'X'Z'\)[/tex] are equal, not doubled.
Thus, the three statements that must be true are:
- [tex]\(\triangle XYZ \cong \triangle X'Y'Z'\)[/tex],
- [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex], and
- [tex]\(XZ = 2X'Z'\)[/tex].
So, the correct options are:
- [tex]\(\triangle X Y Z \cong \triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex]
- [tex]\(\angle X Z Y \cong \angle Y^{\prime} Z^{\prime} X^{\prime}\)[/tex]
- [tex]\(X Z=2 X^{\prime} Z^{\prime}\)[/tex]
