Final answer:
To demonstrate AAS congruence, two angles and a non-included side from one triangle must match those of another. The examples provided explore congruent and similar triangle relationships and their application in physics and astronomy. Determining congruence requires identifying matching angles and sides.
Step-by-step explanation:
To show that two triangles are congruent using the Angle-Angle-Side (AAS) congruence criterion, we must establish that two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle. In the given scenarios, it appears there are discussions about similar triangles, congruent triangles, and relationships based on angles and sides. For example, when the width of the Moon as seen from a point is given as KD = x, and the angles are specified, we can derive that two triangles are congruent if they share two congruent angles and one side. Similarly, in another example, the relationship between triangles based on distances and angles suggests that similar triangles are being analyzed. This analysis could potentially lead to proportionality arguments, which are valuable when discussing congruence or similarity in triangles.
Discussing physical applications, such as in physics, to demonstrate the reliability of mathematical postulates, like the Pythagorean theorem and trigonometry, illustrates the interplay between mathematical concepts and their practical validation through experiments. The questions highlight various mathematical concepts, including congruence, similarity, angle relationships, and proportions. To conclusively answer the initial question about AAS congruence, we would need to identify the relevant angles and sides of the triangles in question and verify their congruency or similarity.