Final answer:
To prove that angles ∠CAB and ∠DBA are congruent, one typically uses principles like the Isosceles Triangle Theorem or triangle congruence postulates. Given the limited information, a precise step-by-step proof can't be provided, but geometric relationships would be key in crafting a proof.
Step-by-step explanation:
The geometry question posed asks us to prove that the angle ∠CAB is congruent to ∠DBA, given that ∠FDC is congruent to ∠FC and ∠DA is congruent to ∠CB. While the specific figure is not provided, based on the information given, one can infer certain geometric properties and relationships that could lead to the desired proof. Typically, one would use principles such as the Isosceles Triangle Theorem, properties of parallel lines, or triangle congruence to establish the necessary congruences.
Without the figure, a step-by-step proof cannot be accurately provided. However, if the triangles mentioned are isosceles for instance, and lines AD and CB intersect at point F, and are parallel, then the alternate interior angles ∠DA and ∠CB would be congruent. Consequently, triangles CAB and DBA would have two pairs of congruent angles, making them similar by the Angle-Angle (AA) similarity postulate. If one can establish that the sides opposite these angles are also congruent, the triangles would be congruent by the Angle-Side-Angle (ASA) postulate, thus proving that ∠CAB and ∠DBA are also congruent.
The given additional information seems unrelated to the initial question and represents various unrelated mathematical concepts. To be precise, there appears to be a mix of questions covering topics such as vector magnitudes and directions, geometric constructions of resultants, and properties of similar triangles. These mathematical snippets do not directly inform the solution to the geometry proof question concerning angles ∠CAB and ∠DBA.