Question
BELOW, In simple term explain the following how you know that each pair of corresponding vertices will overlap perfectly. Then explain why this sequence illustrated the SAS CONGRUENCE THEOREM .
To show that \( \triangle ABC \cong \triangle EFD \) using a sequence of rigid transformations and to demonstrate the SAS (Side-Angle-Side) Congruence Theorem, follow these steps:
### Step 1: Identify Corresponding Parts
First, identify and establish the pairs of corresponding sides and angles. For example, let’s assume:
- \( \overline{AB} \) corresponds to \( \overline{EF} \)
- \( \overline{AC} \) corresponds to \( \overline{ED} \)
- \( \angle A \) corresponds to \( \angle E \)
### Step 2: Rigid Transformations
1. **Translation**:
- Translate \( \triangle ABC \) such that point \( A \) coincides with point \( E \). This initially aligns point \( A \) over point \( E \).
- After this translation, point \( B \) will move to the position that corresponds to point \( F \), and point \( C \) will be positioned in such a way that the sides reflect their length relationships to \( E \), \( F \), and \( D \).
2. **Rotation**:
- Rotate \( \triangle ABC \) around point \( E \) such that \( \overline{AB} \) aligns perfectly with \( \overline{EF} \) and \( \angle A \) aligns with \( \angle E \).
- This rotation will also position \( \overline{AC} \) over \( \overline{ED} \).
3. **Verification of Overlapping Vertices**:
- After the translation and rotation, we will check that \( \overline{AB} \cong \overline{EF} \) is true, indicating that the sides are now congruent as required.
- Since corresponding sides \( \overline{AC} \) and \( \overline{ED} \) are also congruent and the angle \( \angle A \) and \( \angle E \) are congruent by hypothesis (given that triangles are congruent), we’ve satisfied the necessary conditions of the SAS theorem.
### Step 3: Explain Corresponding Vertices Overlapping
After performing the rigid transformations:
- **Vertices Overlap**:
- Point \( A \) perfectly overlaps with point \( E \) from the translation.
- Through the rotation, point \( B \) will now overlap with point \( F \), and point \( C \) will coincide with point \( D \).
- Therefore, the corresponding vertices \( A \), \( B \), and \( C \) will overlap with \( E \), \( F \), and \( D \) respectively.
### Step 4: Illustrating the SAS Congruence Theorem
- By translating \( \triangle ABC \) so that point \( A \) meets point \( E \) and then rotating it, we are ensuring that:
- One pair of sides (\( AB \) and \( EF \)) are congruent.
- The included angle (\( \angle A \) and \( \angle E \)) is congruent.
- The second pair of sides (\( AC \) and \( ED \)) is congruent.
Thus, through a sequence of translations and rotations, we have shown that the relationships among the sides and angles of the triangles meet the criteria of the SAS Congruence Theorem.
### Conclusion
The properties of rigid transformations (translation followed by rotation) ensure that we can confirm \( \triangle ABC \cong \triangle EFD \) and illustrate the SAS theorem through the overlap of corresponding sides and the congruence of included angles.
To show that \( \triangle ABC \cong \triangle EFD \) using a sequence of rigid transformations and to demonstrate the SAS (Side-Angle-Side) Congruence Theorem, follow these steps:
### Step 1: Identify Corresponding Parts
First, identify and establish the pairs of corresponding sides and angles. For example, let’s assume:
- \( \overline{AB} \) corresponds to \( \overline{EF} \)
- \( \overline{AC} \) corresponds to \( \overline{ED} \)
- \( \angle A \) corresponds to \( \angle E \)
### Step 2: Rigid Transformations
1. **Translation**:
- Translate \( \triangle ABC \) such that point \( A \) coincides with point \( E \). This initially aligns point \( A \) over point \( E \).
- After this translation, point \( B \) will move to the position that corresponds to point \( F \), and point \( C \) will be positioned in such a way that the sides reflect their length relationships to \( E \), \( F \), and \( D \).
2. **Rotation**:
- Rotate \( \triangle ABC \) around point \( E \) such that \( \overline{AB} \) aligns perfectly with \( \overline{EF} \) and \( \angle A \) aligns with \( \angle E \).
- This rotation will also position \( \overline{AC} \) over \( \overline{ED} \).
3. **Verification of Overlapping Vertices**:
- After the translation and rotation, we will check that \( \overline{AB} \cong \overline{EF} \) is true, indicating that the sides are now congruent as required.
- Since corresponding sides \( \overline{AC} \) and \( \overline{ED} \) are also congruent and the angle \( \angle A \) and \( \angle E \) are congruent by hypothesis (given that triangles are congruent), we’ve satisfied the necessary conditions of the SAS theorem.
### Step 3: Explain Corresponding Vertices Overlapping
After performing the rigid transformations:
- **Vertices Overlap**:
- Point \( A \) perfectly overlaps with point \( E \) from the translation.
- Through the rotation, point \( B \) will now overlap with point \( F \), and point \( C \) will coincide with point \( D \).
- Therefore, the corresponding vertices \( A \), \( B \), and \( C \) will overlap with \( E \), \( F \), and \( D \) respectively.
### Step 4: Illustrating the SAS Congruence Theorem
- By translating \( \triangle ABC \) so that point \( A \) meets point \( E \) and then rotating it, we are ensuring that:
- One pair of sides (\( AB \) and \( EF \)) are congruent.
- The included angle (\( \angle A \) and \( \angle E \)) is congruent.
- The second pair of sides (\( AC \) and \( ED \)) is congruent.
Thus, through a sequence of translations and rotations, we have shown that the relationships among the sides and angles of the triangles meet the criteria of the SAS Congruence Theorem.
### Conclusion
The properties of rigid transformations (translation followed by rotation) ensure that we can confirm \( \triangle ABC \cong \triangle EFD \) and illustrate the SAS theorem through the overlap of corresponding sides and the congruence of included angles.
Answers
Answered by
GPT-4o mini
To explain how we know that each pair of corresponding vertices will overlap perfectly after the transformations, let's break it down step by step:
### Knowing Corresponding Vertices Overlap
1. **Translation**:
- We start by translating triangle \( ABC \) so that point \( A \) lands directly on point \( E \). This means that after this step, point \( A \) and point \( E \) are exactly at the same location.
2. **Rotation**:
- Next, we perform a rotation around the point \( E \). We rotate triangle \( ABC \) such that line segment \( AB \) aligns perfectly with line segment \( EF \) and angle \( A \) aligns with angle \( E \).
- During this rotation, since we are aligning \( AB \) with \( EF \), point \( B \) will move to the location of point \( F \) because they are the endpoints of those segments, meaning \( B \) will overlap \( F \).
- Likewise, point \( C \) will move to the location of point \( D \) for the same reason.
So, after the translation and rotation, we have:
- Point \( A \) perfectly on \( E \)
- Point \( B \) perfectly on \( F \)
- Point \( C \) perfectly on \( D \)
Thus, we see that each pair of corresponding vertices does overlap perfectly, confirming they are in the same spatial position after the transformations.
### Why this Sequence Illustrates the SAS Congruence Theorem
Now, let’s relate this back to the SAS (Side-Angle-Side) Congruence Theorem. This theorem states that if two triangles have one pair of corresponding sides that are equal in length, the included angles are equal, and another pair of corresponding sides are also equal, then the triangles are congruent.
In our process:
- **First Side Pair**: \( \overline{AB} \cong \overline{EF} \) (the sides we translated and aligned)
- **Included Angle**: \( \angle A \cong \angle E \) (the angle we aligned during rotation)
- **Second Side Pair**: \( \overline{AC} \cong \overline{ED} \) (the sides that are also confirmed congruent)
By fulfilling the requirements of the SAS theorem through our transformations (translation and rotation ensuring the sides and angle match), we demonstrate that \( \triangle ABC \) is congruent to \( \triangle EFD \).
### Conclusion
So, the series of rigid transformations allowed us to verify that the corresponding vertices of the triangles \( ABC \) and \( EFD \) overlap perfectly, satisfying the conditions of the SAS Congruence Theorem, which ensures that the two triangles are congruent.
### Knowing Corresponding Vertices Overlap
1. **Translation**:
- We start by translating triangle \( ABC \) so that point \( A \) lands directly on point \( E \). This means that after this step, point \( A \) and point \( E \) are exactly at the same location.
2. **Rotation**:
- Next, we perform a rotation around the point \( E \). We rotate triangle \( ABC \) such that line segment \( AB \) aligns perfectly with line segment \( EF \) and angle \( A \) aligns with angle \( E \).
- During this rotation, since we are aligning \( AB \) with \( EF \), point \( B \) will move to the location of point \( F \) because they are the endpoints of those segments, meaning \( B \) will overlap \( F \).
- Likewise, point \( C \) will move to the location of point \( D \) for the same reason.
So, after the translation and rotation, we have:
- Point \( A \) perfectly on \( E \)
- Point \( B \) perfectly on \( F \)
- Point \( C \) perfectly on \( D \)
Thus, we see that each pair of corresponding vertices does overlap perfectly, confirming they are in the same spatial position after the transformations.
### Why this Sequence Illustrates the SAS Congruence Theorem
Now, let’s relate this back to the SAS (Side-Angle-Side) Congruence Theorem. This theorem states that if two triangles have one pair of corresponding sides that are equal in length, the included angles are equal, and another pair of corresponding sides are also equal, then the triangles are congruent.
In our process:
- **First Side Pair**: \( \overline{AB} \cong \overline{EF} \) (the sides we translated and aligned)
- **Included Angle**: \( \angle A \cong \angle E \) (the angle we aligned during rotation)
- **Second Side Pair**: \( \overline{AC} \cong \overline{ED} \) (the sides that are also confirmed congruent)
By fulfilling the requirements of the SAS theorem through our transformations (translation and rotation ensuring the sides and angle match), we demonstrate that \( \triangle ABC \) is congruent to \( \triangle EFD \).
### Conclusion
So, the series of rigid transformations allowed us to verify that the corresponding vertices of the triangles \( ABC \) and \( EFD \) overlap perfectly, satisfying the conditions of the SAS Congruence Theorem, which ensures that the two triangles are congruent.