Answer :
To determine which statements are true about the triangles, let's go through each option step-by-step.
### Option 1: [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
A similarity transformation consists of a sequence of rigid motions followed by a dilation. Reflecting and then dilating a triangle does maintain similarity since the shape is preserved even though the size changes. Therefore, [tex]$\triangle XYZ$[/tex] is similar to [tex]$\triangle X'Y'Z'$[/tex].
Verdict: True
### Option 2: [tex]$\angle XZY \approx \angle Y'Z'X'$[/tex]
When a triangle is reflected, the angles do not change; they remain congruent. This property holds irrespective of whether a dilation follows because dilations also do not affect angle measures.
Verdict: True
### Option 3: [tex]$\overline{YX} \approx \overline{Y'X'}$[/tex]
Reflection alone would keep the corresponding side lengths equal, but a subsequent dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] changes the length of all sides in [tex]$\triangle X'Y'Z'$[/tex] to be half the length of the corresponding sides in [tex]$\triangle XYZ$[/tex]. Hence, the lengths are not approximately equal.
Verdict: False
### Option 4: [tex]$XZ = 2 X'Z'$[/tex]
Due to the dilation by a scale factor of [tex]$\frac{1}{2}$[/tex], every side in [tex]$\triangle X'Y'Z'$[/tex], including [tex]$X'Z'$[/tex], will be half the length of the corresponding side in [tex]$\triangle XYZ$[/tex]. Therefore, [tex]$XZ = 2X'Z'$[/tex].
Verdict: True
### Option 5: [tex]$m \angle YXZ = 2 m \angle Y'X'Z'$[/tex]
Reflecting or dilating a triangle does not change the measure of the angles. Since dilations preserve angle measures, the angle measures in [tex]$\triangle XYZ$[/tex] should match those in [tex]$\triangle X'Y'Z'$[/tex] and not be multiplied by a factor of 2.
Verdict: False
In conclusion, the true statements are:
- [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
- [tex]$\angle XZY \approx \angle Y'Z'X'$[/tex]
- [tex]$XZ = 2X'Z'$[/tex]
### Option 1: [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
A similarity transformation consists of a sequence of rigid motions followed by a dilation. Reflecting and then dilating a triangle does maintain similarity since the shape is preserved even though the size changes. Therefore, [tex]$\triangle XYZ$[/tex] is similar to [tex]$\triangle X'Y'Z'$[/tex].
Verdict: True
### Option 2: [tex]$\angle XZY \approx \angle Y'Z'X'$[/tex]
When a triangle is reflected, the angles do not change; they remain congruent. This property holds irrespective of whether a dilation follows because dilations also do not affect angle measures.
Verdict: True
### Option 3: [tex]$\overline{YX} \approx \overline{Y'X'}$[/tex]
Reflection alone would keep the corresponding side lengths equal, but a subsequent dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] changes the length of all sides in [tex]$\triangle X'Y'Z'$[/tex] to be half the length of the corresponding sides in [tex]$\triangle XYZ$[/tex]. Hence, the lengths are not approximately equal.
Verdict: False
### Option 4: [tex]$XZ = 2 X'Z'$[/tex]
Due to the dilation by a scale factor of [tex]$\frac{1}{2}$[/tex], every side in [tex]$\triangle X'Y'Z'$[/tex], including [tex]$X'Z'$[/tex], will be half the length of the corresponding side in [tex]$\triangle XYZ$[/tex]. Therefore, [tex]$XZ = 2X'Z'$[/tex].
Verdict: True
### Option 5: [tex]$m \angle YXZ = 2 m \angle Y'X'Z'$[/tex]
Reflecting or dilating a triangle does not change the measure of the angles. Since dilations preserve angle measures, the angle measures in [tex]$\triangle XYZ$[/tex] should match those in [tex]$\triangle X'Y'Z'$[/tex] and not be multiplied by a factor of 2.
Verdict: False
In conclusion, the true statements are:
- [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
- [tex]$\angle XZY \approx \angle Y'Z'X'$[/tex]
- [tex]$XZ = 2X'Z'$[/tex]
