To show that triangle ABC is the same as triangle DEF (we call this "congruent"), we can use some fun movements called rigid transformations. First, let's think about which parts of the triangles match: angle A matches angle D, angle B matches angle E, and the side connecting angle A and B (which is line AB) matches the side connecting angle D and E (which is line DE). Now, we can move triangle ABC to match triangle DEF. First, we can slide triangle ABC so that point A lands on point D and point B lands on point E. This is called translation. Next, we can spin triangle ABC around point D until angle A lines up perfectly with angle D. After these moves, we see that angle A is equal to angle D, angle B is equal to angle E, and the side AB is the same length as side DE, which helps us use the ASA (Angle-Side-Angle) Congruence Theorem. This tells us that because we showed two angles and their included side are equal, triangle ABC is indeed congruent to triangle DEF!
Describe the following in a paragraph in a 6th graders response
To demonstrate that \( \triangle ABC \cong \triangle DEF \) using a sequence of rigid transformations and to illustrate the ASA (Angle-Side-Angle) Congruence Theorem, you can follow these steps:
### Step 1: Identify Corresponding Parts
Before initiating any transformations, determine how the triangles correspond:
- Assume \( \angle A \) corresponds to \( \angle D \)
- Assume \( \angle B \) corresponds to \( \angle E \)
- The included side \( \overline{AB} \) corresponds to \( \overline{DE} \)
### Step 2: Rigid Transformations
The sequence of rigid transformations that will show \( \triangle ABC \cong \triangle DEF \) could involve the following:
1. **Translation**:
- Translate \( \triangle ABC \) such that point \( A \) coincides with point \( D \) and point \( B \) coincides with point \( E \). This aligns two of the corresponding vertices perfectly, placing \( A \) on \( D \) and \( B \) on \( E \).
2. **Rotation**:
- Rotate \( \triangle ABC \) around point \( D \) such that \( \angle A \) aligns perfectly with \( \angle D \). This rotation will allow us to position all corresponding angles to overlap.
- As a result of this transformation, point \( C \) will move into the position of point \( F \), given that \( \angle B \) aligns with \( \angle E \) after the transformation.
### Step 3: Ensure the Result is Congruent
- **Translation**:
- Translations do not alter the shape or size of the triangle; they merely shift it in space. As such, the resulting figure remains congruent to \( \triangle ABC \).
- **Rotation**:
- Rotations also do not change the size or shape of a figure; they merely rotate it around a specific point. Hence, after the rotation, the angles and sides of \( \triangle ABC \) remain congruent to the corresponding parts of \( \triangle DEF \).
### Step 4: Illustrating the ASA Congruence Theorem
- **Angle-Side-Angle**:
- After completing the above transformations, you will have:
- \( \angle A \) (corresponding to \( \angle D \)) is congruent,
- \( \angle B \) (corresponding to \( \angle E \)) is congruent, and
- The side \( \overline{AB} \) (corresponding to \( \overline{DE} \)) which is between the angles, is congruent.
Thus, we have established two angles and the included side as congruent: this fits the criteria for the ASA theorem.
### Conclusion
Through a sequence of rigid transformations involving translation followed by rotation, the corresponding vertices of \( \triangle ABC \) and \( \triangle DEF \) can be made to overlap perfectly. This validates that \( \triangle ABC \cong \triangle DEF \) and illustrates the ASA Congruence Theorem since we have demonstrated that two angles and the included side are congruent in both triangles.
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