Question

Prove that the criteria for congruent triangles are equivalent to the definition of congruence in terms of rigid motions.

A) SAS (Side-Angle-Side) Congruence Criterion
B) ASA (Angle-Side-Angle) Congruence Criterion
C) SSS (Side-Side-Side) Congruence Criterion
D) AAS (Angle-Angle-Side) Congruence Criterion

Answer 1

The criteria for congruent triangles (SAS, ASA, SSS, AAS) are equivalent to the definition of congruence in terms of rigid motions. This can be proven using rigid motions, such as translation and rotation, to show that triangles with matching sides and angles are congruent.

Step-by-step explanation:

The criteria for congruent triangles (SAS, ASA, SSS, AAS) are equivalent to the definition of congruence in terms of rigid motions.

Let's prove this for the SAS (Side-Angle-Side) criterion as an example:

1. Assume we have two triangles ABC and DEF.
2. If AB = DE, BC = EF, and the included angle ∠B = ∠E, then we can say that triangle ABC is congruent to triangle DEF using the SAS criterion.
3. The proof using rigid motions involves translating and rotating triangle DEF so that side DE coincides with side AB, side EF coincides with side BC, and angle ∠E coincides with angle ∠B.
4. After performing these rigid motions, we would see that triangle DEF perfectly matches triangle ABC. Therefore, they are congruent.

Similar proofs can be done for the other criteria (ASA, SSS, AAS). This shows that the criteria for congruent triangles are equivalent to the definition of congruence in terms of rigid motions.