Final answer:
The term that could be used to prove the triangles congruent in the given context would likely be the Angle-Side-Angle (ASA) congruence criterion, given that we know two angles and the included side are equal for the triangles. Congruence can also be proved using other criteria like Side-Angle-Side (SAS), Side-Side-Side (SSS), or Hypotenuse-Leg (HL) for right triangles, depending on the given information. The Pythagorean theorem can calculate a right triangle's sides.
Step-by-step explanation:
The term used to prove triangle congruence depends on the information given about the triangles. In the provided context, it suggests that the triangles HKD and KFD are congruent. Given that all angles in the shaded triangles are the same, especially the angle of 0.5 degrees at K, and assuming by context we know the sides around these angles are equal, we could use the Angle-Side-Angle (ASA) congruence criterion, which states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
The information also suggests that triangles GFC and AHD are congruent to the shaded triangles, probably based on the same reason. To verify congruence in this case, one would need explicit information on two angles and the included side being equal, or some other congruence criteria being met, such as Side-Angle-Side (SAS), Side-Side-Side (SSS), or Hypotenuse-Leg (HL) for right triangles, among others.
It's also mentioned that applying simple proportions to the geometry mentioned leads to AC being 3R and AB being 3x. Such an approach is crucial in geometric proofs, where the properties and relationships of geometrical figures are used to prove congruence or similarity.
In addition, from an isolated mention is that using the Pythagorean theorem, a fundamental principle in geometry, one could calculate the sides of a right triangle given the other lengths and the right angle. This theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a² + b² = c²).