Answer :
To determine which statements must be 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] to form [tex]$\triangle X' Y' Z'$[/tex], we need to analyze the effects of these transformations on the triangles.
1. Similarity of Triangles:
First, let's consider the similarity of [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X' Y' Z'$[/tex]. Reflection is an isometry, meaning it preserves both angles and side ratios. Dilation by a scale factor, in this case, [tex]$\frac{1}{2}$[/tex], will change the side lengths but not the angles, and the triangles will remain similar because similarity transformations include any dilation. Hence, the triangles [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X' Y' Z'$[/tex] will be similar.
2. Corresponding Angles:
Since the triangles are similar, their corresponding angles will be congruent. So, [tex]$\angle XZY \cong \angle Y'Z'X'$[/tex].
3. Corresponding Sides:
For the sides, since the triangle is dilated by a scale factor of [tex]$\frac{1}{2}$[/tex], each side of [tex]$\triangle XYZ$[/tex] will be half as long in [tex]$\triangle X' Y' Z'$[/tex]. Therefore, we have [tex]$XZ = 2X'Z'$[/tex].
4. Congruency of sides [tex]$YX$[/tex] and [tex]$Y'X'$[/tex]:
After dilation, the length of any side in [tex]$\triangle X' Y' Z'$[/tex], including [tex]$Y'$[/tex] and [tex]$X'$[/tex], should be half of the corresponding side length in [tex]$\triangle XYZ$[/tex]. Hence [tex]$\overline{YX} \cong \overline{Y'X'}$[/tex] is not correct.
5. Angle Measures and Proportionality:
For angles, similar triangles have congruent corresponding angles, not proportional angles. So, [tex]$m \angle YXZ = 2 m \angle Y'X'Z'$[/tex] is incorrect because it suggests an incorrect proportional relationship between angle measures.
To summarize, the true statements about the two triangles are:
1. [tex]$\angle XZY \cong \angle Y'Z'X'$[/tex]
2. [tex]$XZ = 2X'Z'$[/tex]
Therefore, the correct options are 1 and 3.
1. Similarity of Triangles:
First, let's consider the similarity of [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X' Y' Z'$[/tex]. Reflection is an isometry, meaning it preserves both angles and side ratios. Dilation by a scale factor, in this case, [tex]$\frac{1}{2}$[/tex], will change the side lengths but not the angles, and the triangles will remain similar because similarity transformations include any dilation. Hence, the triangles [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X' Y' Z'$[/tex] will be similar.
2. Corresponding Angles:
Since the triangles are similar, their corresponding angles will be congruent. So, [tex]$\angle XZY \cong \angle Y'Z'X'$[/tex].
3. Corresponding Sides:
For the sides, since the triangle is dilated by a scale factor of [tex]$\frac{1}{2}$[/tex], each side of [tex]$\triangle XYZ$[/tex] will be half as long in [tex]$\triangle X' Y' Z'$[/tex]. Therefore, we have [tex]$XZ = 2X'Z'$[/tex].
4. Congruency of sides [tex]$YX$[/tex] and [tex]$Y'X'$[/tex]:
After dilation, the length of any side in [tex]$\triangle X' Y' Z'$[/tex], including [tex]$Y'$[/tex] and [tex]$X'$[/tex], should be half of the corresponding side length in [tex]$\triangle XYZ$[/tex]. Hence [tex]$\overline{YX} \cong \overline{Y'X'}$[/tex] is not correct.
5. Angle Measures and Proportionality:
For angles, similar triangles have congruent corresponding angles, not proportional angles. So, [tex]$m \angle YXZ = 2 m \angle Y'X'Z'$[/tex] is incorrect because it suggests an incorrect proportional relationship between angle measures.
To summarize, the true statements about the two triangles are:
1. [tex]$\angle XZY \cong \angle Y'Z'X'$[/tex]
2. [tex]$XZ = 2X'Z'$[/tex]
Therefore, the correct options are 1 and 3.
