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
EF¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯
Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar
EG¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯
Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar
∠E≅∠X
Responses
∠F≅∠Z
angle upper F congruent to angle upper Z
EF¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯
Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar
EG¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯
Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar
∠E≅∠X
Answers
GPT-4o mini
When a triangle is rotated, the corresponding angles and sides preserve their measurements. For the rotation of triangle \( \triangle EFG \) to create triangle \( \triangle XYZ \) through a -90 degree (clockwise) rotation, the following correspondences can be deduced:
- The point \( E \) will correspond to point \( X \).
- The point \( F \) will correspond to point \( Z \).
- The point \( G \) will correspond to point \( Y \).
Given these correspondences, let's analyze each statement:
1. **\(\angle F \cong \angle Z\)**: This is true, as they correspond under the rotation.
2. **\(EF \cong XZ\)**: This is **not true** since \(EF\) corresponds to side \(ZY\) after rotation.
3. **\(EG \cong XY\)**: This is true, as \(EG\) corresponds to side \(XY\) after the rotation.
4. **\(\angle E \cong \angle X\)**: This is true, as they correspond under the rotation.
From this analysis, the statements that are true after the rotation are:
- \(\angle F \cong \angle Z\)
- \(EG \cong XY\)
- \(\angle E \cong \angle X\)
Thus, the first statement is valid, and the third statement is also valid. The second statement about sides is incorrect.
So the answer is:
\(\angle F \cong \angle Z\), \(EG \cong XY\), and \(\angle E \cong \angle X\) must be true due to the properties of rotation.
- The point \( E \) will correspond to point \( X \).
- The point \( F \) will correspond to point \( Z \).
- The point \( G \) will correspond to point \( Y \).
Given these correspondences, let's analyze each statement:
1. **\(\angle F \cong \angle Z\)**: This is true, as they correspond under the rotation.
2. **\(EF \cong XZ\)**: This is **not true** since \(EF\) corresponds to side \(ZY\) after rotation.
3. **\(EG \cong XY\)**: This is true, as \(EG\) corresponds to side \(XY\) after the rotation.
4. **\(\angle E \cong \angle X\)**: This is true, as they correspond under the rotation.
From this analysis, the statements that are true after the rotation are:
- \(\angle F \cong \angle Z\)
- \(EG \cong XY\)
- \(\angle E \cong \angle X\)
Thus, the first statement is valid, and the third statement is also valid. The second statement about sides is incorrect.
So the answer is:
\(\angle F \cong \angle Z\), \(EG \cong XY\), and \(\angle E \cong \angle X\) must be true due to the properties of rotation.