When a triangle [tex]$\triangle XYZ$[/tex] undergoes a reflection over a vertical line, its shape and size do not change, but its orientation is altered. When it is subsequently dilated by a scale factor of [tex]$\frac{1}{2}$[/tex], its size is reduced to half, but the shape and angles remain the same. Let's analyze each of the given statements to determine which ones must be true:
1. [tex]$\triangle XYZ - \triangle X'Y'Z'$[/tex]:
This statement is offering a comparison in a way that does not describe any geometrical property or relation explicitly. Therefore, it cannot be considered a verifiable geometric statement.
2. [tex]$\angle XZY \cong \angle YZ'X'$[/tex]:
Reflecting a triangle over a vertical line maintains all of its internal angles because reflections are rigid motions and do not affect the measures of the angles. Even after dilating the triangle, the angles remain congruent since dilation alters side lengths but not angles. Hence, this statement is true.
3. [tex]$\overline{YX} \cong \overline{Y'X}$[/tex]:
This statement concerns side lengths and implies congruence. Reflection alone does not alter side lengths, but dilation by [tex]$\frac{1}{2}$[/tex] would halve all side lengths. Therefore, this cannot be true, as [tex]$\overline{YX}$[/tex] would not equal [tex]$\overline{Y'X}$[/tex] after dilation. Hence, this statement is false.
4. [tex]$xz = 2 \times Z'$[/tex]:
Under a dilation with a scale factor of [tex]$\frac{1}{2}$[/tex], the lengths of sides of the triangle are scaled down by half. This implies that the original side [tex]$xz$[/tex] of [tex]$\triangle XYZ$[/tex] will be twice the length of the corresponding side [tex]$x'z'$[/tex] in [tex]$\triangle X'Y'Z'$[/tex]. Thus, this statement must be true.
5. [tex]$m \angle YXZ = 2 m \angle Y'X'Z'$[/tex]:
As mentioned earlier, reflections and dilations do not alter the measures of the angles of a triangle. The angles remain the same before and after these transformations. Therefore, the measure of [tex]$\angle YXZ$[/tex] cannot be twice the measure of [tex]$\angle Y'X'Z'$[/tex]. This statement must be false.
From the above analysis, the three statements that must be true are:
- [tex]$\angle XZY \cong \angle YZ'X'$[/tex]
- [tex]$xz = 2 \times Z'$[/tex]
Hence, the correct choices are:
- [tex]$\angle XZY \cong \angle YZ'X'$[/tex]
- [tex]$xz = 2 \times Z'$[/tex]
The third verifiable statement, as shown in the analysis, does not exist among the provided options in a descriptive manner. The choices that must be true from the given options are [tex]$\boxed{\angle XZY \cong \angle YZ'X'}$[/tex] and [tex]$\boxed{xz = 2 \times Z'}$[/tex].
