To show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle DEF \), we can use a sequence of rigid transformations—specifically reflections, rotations, or translations—since these transformations preserve the lengths of sides and the measures of angles, thereby resulting in congruent figures.
Sequence of Rigid Transformations
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Reflection:
- We can begin by reflecting \( \triangle ABC \) across a line that bisects angle \( A \) (the angle between sides \( AB \) and \( AC \)).
- This reflection will ensure that angle \( A \) aligns with angle \( D \) (as \( \angle A \cong \angle D \)) and angle \( C \) aligns with angle \( F \) (as \( \angle C \cong \angle F \)).
- After the reflection, we will have \( A'B'C' \) where \( A' \) aligns with \( D \), and both \( B' \) and \( C' \) are positioned correctly relative to angles \( D \) and \( F \) respectively.
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Rotation:
- Next, we can apply a rotation around point \( D \) to position point \( A' \) so that it overlaps with point \( D \).
- The rotation will ensure that the angle positions maintain equality due to the preservation of angles and distances.
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Translation:
- Finally, we can translate \( \triangle A'B'C' \) to align \( B' \) with \( E \) and \( C' \) with \( F \) while ensuring \( A' \) is already positioned over \( D \).
- Such a translation will maintain the lengths of sides \( AC \) and \( DF \) as congruent segments (as given).
Why Each Transformation Maintains Congruence
- Reflection: This operation flips the triangle over a line, which maintains all side lengths and angle measures.
- Rotation: This movement turns the triangle around a point, keeping all distances and angles intact, and thus preserves the congruence of the figure.
- Translation: This transformation slides the triangle without altering its shape or size; consequently, all lengths and angles will remain the same.
Illustration of the ASA Congruence Theorem
The ASA (Angle-Side-Angle) Congruence Theorem states that if in two triangles, two angles and the included side of one triangle are respectively congruent to two angles and the included side of the other triangle, then the triangles are congruent.
- In the transformations described, we start off knowing:
- \( \angle A \cong \angle D \)
- \( \angle C \cong \angle F \)
- \( AC \cong DF \) (the included side).
These congruent angles and the included side (which has been preserved through reflections, rotations, and translations) satisfy the conditions of the ASA theorem. Thus, any sequence of rigid transformations applied to \( \triangle ABC \) demonstrates that \( \triangle DEF \) can be effectively created, thereby confirming \( \triangle ABC \cong \triangle DEF \).
In conclusion, by performing these rigid transformations and verifying the preserved relationships among the angles and sides, we can confidently assert that \( \triangle ABC \equiv \triangle DEF \) as illustrated by the ASA Congruence Theorem.
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