Question

Describe a sequence of rigid transformations that could be used to show that △ABC≅△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

Without a picture I’ll describe a general, standard sequence of rigid motions (isometries) that sends triangle ABC onto triangle DEF, assuming you are given ∠A ≅ ∠D, ∠B ≅ ∠E, and the included side AB ≅ DE (the ASA hypothesis).

1. Translate triangle ABC so that A moves to D.
- A translation moves every point the same vector. It preserves all distances and angles, so the translated triangle is congruent to the original.

2. Rotate the translated triangle about point D until the image of ray AB lies on ray DE (equivalently, until the image of B falls on E).
- A rotation about D preserves distances and angles, so the image of segment AB remains the same length as AB (which equals DE). Because AB and DE have the same length, and we have rotated so the ray through B lines up with the ray through E, the image of B lands exactly on E. Thus after this rotation the image of the segment AB coincides with DE.

3. (If needed) Reflect across the line DE.
- In some cases the orientation will already match and no reflection is needed; in other cases a reflection across the line DE is required to get the correct “side” of DE. Reflection is a rigid motion (preserves distances and angles), so congruence is preserved.

Why these motions give congruent figures
- Each step is a rigid transformation: translations, rotations and reflections preserve distances and angles. Composing rigid motions produces another rigid motion. So the image triangle after the sequence has exactly the same side lengths and angles as the original triangle — i.e., it is congruent.

Why this sequence illustrates ASA
- After steps 1–2 (or after composing a translation and rotation/reflection), AB is placed exactly on DE and A corresponds to D, B to E. Because ∠A ≅ ∠D and ∠B ≅ ∠E, the rays through A and B that determine the position of the third vertex C map to the rays through D and E that determine F. The intersection of those two rays with the given segment AB (now DE) is unique, so the image of C must coincide with F. Thus the whole triangle maps onto triangle DEF. This demonstrates the ASA principle: two angles and the included side fix the triangle up to rigid motion, so triangles with two corresponding angles and the included side equal are congruent.