Final answer:
To prove Theorem 4-4 (CPCTC), we use rigid motion to overlap congruent triangles ABC and XYZ, ensuring all corresponding parts match. This process involves translation, rotation, and possibly reflection. This theorem is also useful in physics-related scenarios such as uniform circular motion or simple harmonic motion.
Step-by-step explanation:
To prove theorem 4-4, which states that corresponding parts of congruent triangles are congruent (CPCTC), we can use rigid motion. Rigid motion includes transformations such as translation, rotation, and reflection that do not change the size and shape of a figure. Given that triangles ABC and XYZ are congruent with ABC = XYZ, we can apply a rigid motion to position triangle ABC onto triangle XYZ in such a way that every corresponding side and angle matches perfectly.
First, we can translate triangle ABC so that point A aligns with point X. Next, we can rotate the triangle around point X until side AB aligns with side XY. If necessary, we might apply a reflection to ensure that point B falls onto point Y. Once all vertices are matched (A with X, B with Y, and C with Z), since the triangles are congruent, all corresponding sides and angles are congruent by definition.
As an example of the application of the theorem, consider the triangles formed in scenarios involving uniform circular motion or simple harmonic motion, as mentioned in various figures. In such cases, the congruence of triangles and the properties of CPCTC can play a crucial role in solving for unknowns or proving specific properties like invariants under rotations or the application of the Pythagorean theorem a² + b² = c².