To determine which statements must be true for the given transformations, let's consider the properties of reflection and dilation.
1. Similarity of triangles:
- Reflections and dilations both preserve the shape of a triangle, retaining its angles.
- After reflecting [tex]$\triangle XYZ$[/tex], the new triangle is congruent to the original in terms of angle measures but might be positioned differently.
- Dilation with a scale factor of [tex]$\frac{1}{2}$[/tex] changes the size of the triangle but keeps the angles the same and side lengths proportional.
- Therefore, [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] are similar triangles because their corresponding angles are equal and the side lengths are proportional.
[tex]\[
\boxed{\triangle XYZ \sim \triangle X'Y'Z'}
\][/tex]
2. Angle congruence:
- Reflecting a triangle over a vertical line does not change the measures of the angles; it only changes the orientation.
- Thus, the angles of [tex]$\triangle XYZ$[/tex] are congruent to the corresponding angles of [tex]$\triangle X'Y'Z'$[/tex].
- Specifically, [tex]$\angle XZY$[/tex] from [tex]$\triangle XYZ$[/tex] will be congruent to [tex]$\angle Y'Z'X'$[/tex] from [tex]$\triangle X'Y'Z'$[/tex] after reflection.
[tex]\[
\boxed{\angle XZY \cong \angle Y'Z'X'}
\][/tex]
3. Congruence of sides:
- After reflecting [tex]$\triangle XYZ$[/tex], the lengths of the sides do not change.
- Dilation with a scale factor of [tex]$\frac{1}{2}$[/tex] means the new triangle's sides are half the length of the original triangle’s sides.
- Therefore, the side lengths in [tex]$\triangle X'Y'Z'$[/tex] are not congruent to the original triangle's sides but are scaled down by a factor of [tex]$\frac{1}{2}$[/tex].
[tex]\[
\not\cong \boxed{\overline{YX} \not\cong \overline{Y'X'}}
\][/tex]
4. Proportionality of sides:
- If [tex]$XZ$[/tex] is a side in the original triangle [tex]$\triangle XYZ$[/tex] and [tex]$X'Z'$[/tex] is the corresponding side in the dilated triangle, then the length of [tex]$XZ$[/tex] will be twice the length of [tex]$X'Z'$[/tex] because the dilation factor is [tex]$\frac{1}{2}$[/tex].
- Thus, [tex]$XZ = 2 \cdot X'Z'$[/tex].
[tex]\[
\boxed{XZ = 2 \cdot X'Z'}
\][/tex]
5. Angle measure doubling:
- Dilation changes the side lengths of the triangle but does not alter the measures of the angles.
- Therefore, [tex]$m \angle YXZ$[/tex] is equal to [tex]$m \angle Y'X'Z'$[/tex], and it is not doubled.
[tex]\[
\not\boxed{m \angle YXZ \not= 2 \cdot m \angle Y'X'Z'}
\][/tex]
The three correct statements that must be true based on the transformations of [tex]$\triangle XYZ$[/tex] to [tex]$\triangle X'Y'Z'$[/tex] are:
1. [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex].
2. [tex]$\angle XZY \cong \angle Y'Z'X'$[/tex].
4. [tex]$XZ = 2 \cdot X'Z'$[/tex].
