Describe a sequence of rigid transformations that could be used to show that triangle ABC ~= triangle DEF. 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

Answer 1

To show that triangle ABC is congruent to triangle DEF (\( \triangle ABC \cong \triangle DEF \)) using rigid transformations, we can employ a specific sequence of transformations: translation, rotation, and possibly reflection. Here’s how we can structure these transformations, along with an explanation of how each step maintains congruence and how it relates to the Angle-Side-Angle (ASA) Congruence Theorem.

Sequence of Rigid Transformations

1. Translation:

   • We can translate triangle ABC so that one of its vertices (let's say point A) coincides with the corresponding vertex in triangle DEF (point D).
   • Reason: Translation is a rigid transformation; it shifts the entire figure without altering its size or shape, hence preserving congruence.

2. Rotation:

   • Next, we can rotate triangle ABC around the point D (the vertex now aligned with vertex D of triangle DEF) so that the side AB aligns with side DE.
   • Reason: Rotation is another rigid transformation that maintains the lengths of sides and measures of angles, thus keeping the triangles congruent.

3. Reflection (if needed):

   • If upon rotation, triangle ABC does not perfectly overlap triangle DEF, we can apply a reflection across the line of symmetry between the two triangles. For example, if the orientation is incorrect after rotation, reflecting across the line DE will ensure proper alignment.
   • Reason: Reflection is also a rigid transformation that preserves distance and angle measures.

Explanation of Congruent Figures

Each of these transformations—translation, rotation, and reflection—are classified as rigid transformations. Rigid transformations preserve the distances between points, angles, and overall shape of the figure. Therefore, by applying these transformations, we can conclude that triangle ABC will be congruent to triangle DEF since no dimension or shape has been altered.

Illustration of the ASA Congruence Theorem

The ASA (Angle-Side-Angle) Congruence Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

• When we translated triangle ABC such that point A coincides with point D, we aligned two corresponding angles (angle A with angle D).
• During the rotation stage, while aligning side AB with side DE, we are effectively keeping the included angle (angle A between sides AB and AC) congruent with the corresponding angle in triangle DEF (angle D between sides DE and DF).
• The fact that we aligned side AB with side DE confirms that we also preserve the length of the included side.

By successfully demonstrating these transformations and preserving the appropriate angles and sides, we can conclude that \( \triangle ABC \cong \triangle DEF \), effectively illustrating the ASA Congruence Theorem.