Certainly! Let's break down the problem and determine which of the given statements are true when [tex]$\triangle XYZ$[/tex] is reflected over a vertical line and then dilated by a scale factor of [tex]\(\frac{1}{2}\)[/tex], resulting in [tex]\(\triangle X'Y'Z'\)[/tex].
1. Similarity of Triangles:
[tex]\[
\triangle XYZ \sim \triangle X'Y'Z'
\][/tex]
When a triangle is reflected and then scaled by any multiplier, the shape of the triangle does not change; only its size does. Reflection and dilation preserve the angles and the ratios of the sides. Therefore, [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex] are similar triangles.
[tex]\[
\text{True}
\][/tex]
2. Angle Congruence:
[tex]\[
\angle XZY \cong \angle Y'Z'X'
\][/tex]
Reflection over a line and dilation by a scale factor do not affect the measures of the angles of the triangle. Therefore, corresponding angles remain congruent.
[tex]\[
\text{True}
\][/tex]
3. Approximation of segments:
[tex]\[
\overline{YX} \approx \overline{Y'X}
\][/tex]
Since the transformation involves a dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex] and not an exact congruence transformation (where lengths would remain the same), this statement about the lengths being approximately equal is incorrect.
[tex]\[
\text{False}
\][/tex]
4. Scale of Side Lengths:
[tex]\[
XZ = 2X'Z'
\][/tex]
Given the scale factor of [tex]\(\frac{1}{2}\)[/tex], the sides of [tex]\(\triangle X'Y'Z'\)[/tex] are half the length of the corresponding sides in [tex]\(\triangle XYZ\)[/tex]. Hence, [tex]\(XZ\)[/tex] would be twice the length of [tex]\(X'Z'\)[/tex].
[tex]\[
\text{True}
\][/tex]
5. Scaling Angle Measures:
[tex]\[
m \angle YXZ=2 m \angle YY'X'Z'
\][/tex]
Angle measures are invariant under reflection and dilation. This means that if the measure of an angle in [tex]\(\triangle XYZ\)[/tex] is [tex]\(m\angle YXZ\)[/tex], it is the same as the corresponding angle in [tex]\(\triangle X'Y'Z'\)[/tex]. They do not double or halve.
[tex]\[
\text{False}
\][/tex]
So, the three true statements are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]
4. [tex]\(XZ = 2 X'Z'\)[/tex]
Hence, the results are:
[tex]\[
\boxed{(\text{True}, \text{True}, \text{False}, \text{True}, \text{False})}
\][/tex]
