Question
△EFG is rotated −90 degrees (clockwise) to create △XYZ. Which of the following statements must be true?
Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar
Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar
∠F≅∠Z
∠E≅∠X
Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar
Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar
∠F≅∠Z
∠E≅∠X
Answers
Answered by
GPT-4o mini
When triangle \( \triangle EFG \) is rotated \(-90\) degrees (clockwise) to create triangle \( \triangle XYZ \), several properties and relationships between the triangles can be established due to the nature of rotation.
1. **Congruence of Sides**: Rotations preserve distance, so the corresponding sides of the triangles will be congruent. Thus:
- \( \overline{EF} \cong \overline{XY} \)
- \( \overline{FG} \cong \overline{YZ} \)
- \( \overline{EG} \cong \overline{XZ} \)
2. **Congruence of Angles**: Rotations also preserve angles and their measures. Therefore, corresponding angles in the triangles will also be congruent:
- \( \angle E \cong \angle X \)
- \( \angle F \cong \angle Y \)
- \( \angle G \cong \angle Z \)
Given this information, we can analyze the provided statements:
1. **\( \overline{EF} \cong \overline{XZ} \)** - This is false, as these segments are not corresponding; \( \overline{EF} \) corresponds to \( \overline{XY} \).
2. **\( \overline{EG} \cong \overline{XY} \)** - This is false; \( \overline{EG} \) corresponds to \( \overline{XZ} \).
3. **\( \angle F \cong \angle Z \)** - This statement is true based on the property of rotation conserving angle measures.
4. **\( \angle E \cong \angle X \)** - This statement is also true, as corresponding angles remain congruent under rotation.
From this analysis, the statements that must be true are:
- \( \angle F \cong \angle Z \)
- \( \angle E \cong \angle X \)
In conclusion, the true statements after the rotation of \( \triangle EFG \) to \( \triangle XYZ \) are:
- \( \angle E \cong \angle X \)
- \( \angle F \cong \angle Z \)
1. **Congruence of Sides**: Rotations preserve distance, so the corresponding sides of the triangles will be congruent. Thus:
- \( \overline{EF} \cong \overline{XY} \)
- \( \overline{FG} \cong \overline{YZ} \)
- \( \overline{EG} \cong \overline{XZ} \)
2. **Congruence of Angles**: Rotations also preserve angles and their measures. Therefore, corresponding angles in the triangles will also be congruent:
- \( \angle E \cong \angle X \)
- \( \angle F \cong \angle Y \)
- \( \angle G \cong \angle Z \)
Given this information, we can analyze the provided statements:
1. **\( \overline{EF} \cong \overline{XZ} \)** - This is false, as these segments are not corresponding; \( \overline{EF} \) corresponds to \( \overline{XY} \).
2. **\( \overline{EG} \cong \overline{XY} \)** - This is false; \( \overline{EG} \) corresponds to \( \overline{XZ} \).
3. **\( \angle F \cong \angle Z \)** - This statement is true based on the property of rotation conserving angle measures.
4. **\( \angle E \cong \angle X \)** - This statement is also true, as corresponding angles remain congruent under rotation.
From this analysis, the statements that must be true are:
- \( \angle F \cong \angle Z \)
- \( \angle E \cong \angle X \)
In conclusion, the true statements after the rotation of \( \triangle EFG \) to \( \triangle XYZ \) are:
- \( \angle E \cong \angle X \)
- \( \angle F \cong \angle Z \)