Question
describe a sequence of rigid transformations that could be used to show that triangle ABC = triangle EFD. Be sure to explain how you know that each pair of illustrates the SAS Congruence Theorem. Keep simple and short.
Answers
GPT-4o mini
To show that triangle ABC is congruent to triangle EFD using rigid transformations and the SAS Congruence Theorem, you can follow these steps:
1. **Translation**: Move triangle ABC so that point A coincides with point E. This keeps the sides and angles of the triangle unchanged.
2. **Rotation**: Rotate triangle ABC around point E until side AB aligns perfectly with side EF. This will maintain the length of the sides and the angles.
3. **Reflection (if necessary)**: If the orientation of triangle ABC does not match triangle EFD after rotation, reflect triangle ABC over the line EF. This will ensure that both triangles have the same orientation.
Now, let’s confirm SAS:
- **Side AB and EF**: After translation, we know AB is congruent to EF because they correspond to each other.
- **Side AC and ED**: After the transformations, if we can show AC is congruent to ED at this stage, we have the second side pair.
- **Angle ∠A and ∠E**: The angle formed at point E (from the translation and rotation) is congruent to angle ∠A.
Since we have established two pairs of corresponding sides (AB = EF and AC = ED) and the angle between them (∠A = ∠E), we can conclude that triangle ABC is congruent to triangle EFD by the SAS Congruence Theorem.
1. **Translation**: Move triangle ABC so that point A coincides with point E. This keeps the sides and angles of the triangle unchanged.
2. **Rotation**: Rotate triangle ABC around point E until side AB aligns perfectly with side EF. This will maintain the length of the sides and the angles.
3. **Reflection (if necessary)**: If the orientation of triangle ABC does not match triangle EFD after rotation, reflect triangle ABC over the line EF. This will ensure that both triangles have the same orientation.
Now, let’s confirm SAS:
- **Side AB and EF**: After translation, we know AB is congruent to EF because they correspond to each other.
- **Side AC and ED**: After the transformations, if we can show AC is congruent to ED at this stage, we have the second side pair.
- **Angle ∠A and ∠E**: The angle formed at point E (from the translation and rotation) is congruent to angle ∠A.
Since we have established two pairs of corresponding sides (AB = EF and AC = ED) and the angle between them (∠A = ∠E), we can conclude that triangle ABC is congruent to triangle EFD by the SAS Congruence Theorem.