To prove the congruence of the given triangles, one would likely use the SAS or ASA Postulates, backed by trigonometry to calculate and compare corresponding angles and sides, ensuring that each triangle matches with regard to layout on a plane.
To prove that the triangles mentioned are congruent, one would need to use a method that compares their sides and angles.
Considering the provided details, it seems that these triangles would likely be proven congruent using either the Side-Angle-Side (SAS) Postulate or the Angle-Side-Angle (ASA) Postulate.
Both of these postulates require two sides and the included angle to be congruent (SAS) or two angles and the included side (ASA) to establish congruence between two triangles.
Given that the width of the Moon (KD = x) and the angle (KHD = 0.5 degrees) is the same in both shaded triangles HKD and KFD, and by extending the line AD = R to F making AC = 3R, one could determine congruence by showing that corresponding sides and angles are equal.
Trigonometry, which is reliable due to its logical derivations from a set of postulates, may also be used to assist in proving the congruence of triangles when dealing with right triangles or when calculating angles and sides.
We know that if the triangles are situated on the same plane and have their sides and angles matching respectively, then we can conclude that they are congruent.