Final answer:
The question requires understanding geometry to verify congruence of trapezoids through transformations. Transformations that preserve size and shape, such as rotations, can prove congruence. Invariant distances under rotation can be demonstrated with the Pythagorean Theorem and trigonometric principles.
Step-by-step explanation:
The question presented falls within the realm of Mathematics, specifically within the study of geometry and its application in verifying congruences through transformations. When determining if two shapes are congruent, one must show that all corresponding sides and angles are equivalent. While the provided question seems to be missing specific transformation sequences, under general circumstances, sequences such as translations, rotations, and reflections can indeed show congruence if they correctly map one figure onto the other without altering size or shape. To demonstrate that the distance between points remains the same under rotations (an aspect of proving congruence), one could use the Pythagorean Theorem and trigonometric principles, since these mathematical tools assure us that distances and angles are preserved during such transformations.
For instance, if we are to show that the distance between points P and Q is invariant under rotation, we can apply the concept that rotating a plane does not change the distance between any two points within it. The distance formula, derived from the Pythagorean Theorem, can confirm this: If P is at (x1, y1) and Q is at (x2, y2), the distance between them is √[(x2 - x1)2 + (y2 - y1)2]. Under rotation, although the individual coordinates of points P and Q might change, the overall distance calculated from this formula would remain constant.