# Week 8 Day 1 Opener

Proof: 1. Segments AB and DE are the same length so they are congruent. Therefore, there is a rigid motion that takes AB to DE. 2. Apply that rigid motion to triangle ABC. The image of A will coincide with D, and the image of B will coincide with E. 3. We cannot be sure that the image of C coincides with F yet. If necessary, reflect the image of triangle ABC across DE to be sure the image of C, which we will call C', is on the same side of DE as F. (This reflection does not change the image of A or B.) 4. We know the image of angle A is congruent to angle D because rigid motions don’t change the size of angles. 5.  C' must be on ray DF since both C' and F are on the same side of DE, and make the same angle with it at D. 6. Segment DC' is the image of AC and rigid motions preserve distance, so they must have the same length. 7. We also know AC has the same length as DF. So DC' and DF must be the same length. 8. Since C' and F are the same distance along the same ray from D, they have to be in the same place. 9. We have shown that a rigid motion takes A to D, B to E, and C to F; therefore, triangle ABC is congruent to triangle DEF.