Question
Describe a sequence of rigid transformations that could be used to show that △ABC≅△DEF
△
𝐴
𝐵
𝐶
≅
△
𝐷
𝐸
𝐹
. Explain how you know each transformation will result in a congruent figure. Then, explain how this sequence of rigid transformations illustrates the ASA Congruence Theorem.
(4 points)
△
𝐴
𝐵
𝐶
≅
△
𝐷
𝐸
𝐹
. Explain how you know each transformation will result in a congruent figure. Then, explain how this sequence of rigid transformations illustrates the ASA Congruence Theorem.
(4 points)
Answers
GPT-4o mini
To demonstrate that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent (\( \triangle ABC \cong \triangle DEF \)), we can use a sequence of rigid transformations. Rigid transformations include translations, rotations, and reflections; these do not alter the size or shape of a figure, preserving congruence.
### Sequence of Rigid Transformations:
1. **Translation**: Start by translating \( \triangle ABC \) so that point \( A \) coincides with point \( D \). Translation moves every point the same distance in the same direction, ensuring that the shape and orientation of \( \triangle ABC \) are preserved.
2. **Rotation**: Next, rotate \( \triangle ABC \) around point \( D \) (which is now point \( A \)) until point \( B \) aligns with point \( E \). Rotation preserves the size and shape of the triangle, and therefore, the congruence remains intact.
3. **Reflection (if necessary)**: If after the rotation point \( C \) does not align with point \( F \), perform a reflection over the line that bisects the angle formed by points \( D \) and \( E \). Reflection is also a rigid transformation that maintains congruence.
### Explanation of Congruence:
Each transformation in this sequence maintains the congruence of the triangles because they are all rigid transformations. Specifically:
- **Translation** does not change the distance between points; thus, the lengths of sides and measures of angles remain unchanged.
- **Rotation** similarly keeps all the distances unchanged, rotating the figure but not altering its dimensions or angles.
- **Reflection** maintains congruence as well; it mirrors the shape without changing side lengths or angle measures.
### ASA Congruence Theorem:
This sequence of transformations illustrates the **Angle-Side-Angle (ASA) Congruence Theorem**. The ASA theorem states that if two angles and the side between them in one triangle are congruent to two angles and the side between them in another triangle, then the triangles are congruent.
In this scenario:
- The transformation shows that we can align triangle \( ABC \) to triangle \( DEF \) such that:
- \( \angle A \cong \angle D \)
- \( \angle B \cong \angle E \)
- The side \( AB \) (the side between those angles) is congruent to side \( DE \).
Thus, by utilizing these rigid transformations and verifying congruence between the angles and the included side, we conclude that \( \triangle ABC \cong \triangle DEF \) according to the ASA Congruence Theorem.
### Sequence of Rigid Transformations:
1. **Translation**: Start by translating \( \triangle ABC \) so that point \( A \) coincides with point \( D \). Translation moves every point the same distance in the same direction, ensuring that the shape and orientation of \( \triangle ABC \) are preserved.
2. **Rotation**: Next, rotate \( \triangle ABC \) around point \( D \) (which is now point \( A \)) until point \( B \) aligns with point \( E \). Rotation preserves the size and shape of the triangle, and therefore, the congruence remains intact.
3. **Reflection (if necessary)**: If after the rotation point \( C \) does not align with point \( F \), perform a reflection over the line that bisects the angle formed by points \( D \) and \( E \). Reflection is also a rigid transformation that maintains congruence.
### Explanation of Congruence:
Each transformation in this sequence maintains the congruence of the triangles because they are all rigid transformations. Specifically:
- **Translation** does not change the distance between points; thus, the lengths of sides and measures of angles remain unchanged.
- **Rotation** similarly keeps all the distances unchanged, rotating the figure but not altering its dimensions or angles.
- **Reflection** maintains congruence as well; it mirrors the shape without changing side lengths or angle measures.
### ASA Congruence Theorem:
This sequence of transformations illustrates the **Angle-Side-Angle (ASA) Congruence Theorem**. The ASA theorem states that if two angles and the side between them in one triangle are congruent to two angles and the side between them in another triangle, then the triangles are congruent.
In this scenario:
- The transformation shows that we can align triangle \( ABC \) to triangle \( DEF \) such that:
- \( \angle A \cong \angle D \)
- \( \angle B \cong \angle E \)
- The side \( AB \) (the side between those angles) is congruent to side \( DE \).
Thus, by utilizing these rigid transformations and verifying congruence between the angles and the included side, we conclude that \( \triangle ABC \cong \triangle DEF \) according to the ASA Congruence Theorem.