Question

11 activity 1 synthesis: proving the side-angle-side triangle congruence theorem
lets share our proofs.

• what do the sequences of transformations have in common? how are they different?
• what can we add or change in our existing proof to account for the reflection?

Explanation:

1. For transformation sequences:

• Commonality: All sequences of rigid transformations (translations, rotations, reflections) used to prove SAS congruence preserve side lengths and angle measures, which is required to map one triangle onto the other to show congruence.
• Difference: Sequences may vary by the order of transformations (e.g., translate first then rotate vs. rotate first then translate) or whether a reflection is included. Reflections create a mirror image, while translations/rotations map a triangle to a congruent triangle in the same orientation.

1. For accounting for reflection in proofs:

To include reflection in an existing SAS proof, explicitly state that a reflection is a rigid transformation that preserves side lengths and angle measures. If the original proof only uses translations and rotations (which preserve orientation), note that a reflection is needed when the two triangles are mirror images (opposite orientations). You can add a step that reflects one triangle over a line (e.g., the line containing the included side) to align its orientation with the second triangle, then proceed with the original transformation steps (translation/rotation) to map it onto the second triangle.

Answer:

1. Commonality: All preserve side/angle measures.

Difference: Order of steps or inclusion of reflection (which changes orientation, while translations/rotations do not).

1. Explicitly state reflection is a rigid transformation that preserves congruence; add a step to reflect one triangle to match the orientation of the other, then use existing transformation steps to map the reflected triangle onto the second triangle.