Question
△EFG
is rotated −90
degrees (clockwise) to create △XYZ
. Which of the following statements must be true?(1 point)
Responses
∠F≅∠Z
angle upper F congruent to angle upper Z
EG¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯
Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar
EF¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯
Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar
∠E≅∠X
is rotated −90
degrees (clockwise) to create △XYZ
. Which of the following statements must be true?(1 point)
Responses
∠F≅∠Z
angle upper F congruent to angle upper Z
EG¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯
Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar
EF¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯
Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar
∠E≅∠X
Answers
GPT-4o mini
When triangle \( \triangle EFG \) is rotated \( -90 \) degrees (clockwise) to create triangle \( \triangle XYZ \), we need to analyze how the angles and sides correspond after the rotation.
1. **Angle Correspondence**:
- In a \( -90 \) degree rotation, the original points shift positions:
- If \( E \) goes to \( X \), \( F \) goes to \( Y \), and \( G \) goes to \( Z \), the angle correspondence would be:
- \( \angle E \) (original angle at \( E \)) corresponds to \( \angle X \).
- \( \angle F \) corresponds to \( \angle Y \).
- \( \angle G \) corresponds to \( \angle Z \).
- Therefore, \( \angle E \) is congruent to \( \angle X \).
2. **Side Correspondence**:
- The sides will correspond based on their positions:
- \( EF \) becomes \( XY \).
- \( FG \) becomes \( YZ \).
- \( EG \) becomes \( XZ \).
**Analyzing each statement:**
- \( \angle F \cong \angle Z \): **False** (According to the rotation, \( \angle F \) corresponds to \( \angle Y \), not \( \angle Z \)).
- \( \overline{EG} \cong \overline{XY} \): **False** (According to the rotation, \( \overline{EG} \) corresponds to \( \overline{XZ} \)).
- \( \overline{EF} \cong \overline{XZ} \): **False** (According to the rotation, \( \overline{EF} \) corresponds to \( \overline{XY} \)).
- \( \angle E \cong \angle X \): **True** (This matches the correspondence of angles after rotation).
**Conclusion**: The only statement that must be true after the rotation is:
- \( \angle E \cong \angle X \).
1. **Angle Correspondence**:
- In a \( -90 \) degree rotation, the original points shift positions:
- If \( E \) goes to \( X \), \( F \) goes to \( Y \), and \( G \) goes to \( Z \), the angle correspondence would be:
- \( \angle E \) (original angle at \( E \)) corresponds to \( \angle X \).
- \( \angle F \) corresponds to \( \angle Y \).
- \( \angle G \) corresponds to \( \angle Z \).
- Therefore, \( \angle E \) is congruent to \( \angle X \).
2. **Side Correspondence**:
- The sides will correspond based on their positions:
- \( EF \) becomes \( XY \).
- \( FG \) becomes \( YZ \).
- \( EG \) becomes \( XZ \).
**Analyzing each statement:**
- \( \angle F \cong \angle Z \): **False** (According to the rotation, \( \angle F \) corresponds to \( \angle Y \), not \( \angle Z \)).
- \( \overline{EG} \cong \overline{XY} \): **False** (According to the rotation, \( \overline{EG} \) corresponds to \( \overline{XZ} \)).
- \( \overline{EF} \cong \overline{XZ} \): **False** (According to the rotation, \( \overline{EF} \) corresponds to \( \overline{XY} \)).
- \( \angle E \cong \angle X \): **True** (This matches the correspondence of angles after rotation).
**Conclusion**: The only statement that must be true after the rotation is:
- \( \angle E \cong \angle X \).