Final answer:
The ASA Congruence for triangles △PQR and △STU is proven by verifying that two angles and the included side are congruent in both triangles. This corroborates that the third angles and remaining sides will also be congruent due to the angle sum in a triangle being 180 degrees, thus the two triangles are congruent.
Step-by-step explanation:
To prove the Angle-Side-Angle (ASA) Congruence for triangles △PQR and △STU in a Hilbert plane, let's first establish our givens:
- PQ ≅ ST (the sides are congruent)
- ∠RPQ≅∠UST (the angles are congruent)
- ∠PQR≅∠STU (the angles are congruent)
In a Hilbert plane, similar to Euclidean geometry, the sum of angles in a triangle is 180 degrees. By the given information, we can say that both triangles △PQR and △STU have two angles equal and one side enclosed by these angles is also equal. This satisfies the criteria for ASA Congruence, which states that if two angles and the included side of one triangle are the same as two angles and the included side of another triangle, then the two triangles are congruent.
Since ∠RPQ≅∠UST and ∠PQR≅∠STU, it implies that the third angle (∠PRQ and ∠SUT, respectively) must also be congruent because the total angle sum for both triangles must be 180 degrees. Coupled with the fact that PQ ≅ ST, triangle △PQR is congruent to triangle △STU by the ASA Postulate.
This proof is a fundamental aspect of plane geometry and assures that when triangles satisfy ASA conditions, their remaining elements (sides and angles) must also be congruent, establishing their overall congruence.