Final answer:
To prove triangle congruence, the ASA, AAS, or SSS theorem can be used, depending on which sides and angles are known to be congruent between the triangles. Without specific information on the triangles, it's not possible to choose the correct theorem. Other parts of the question refer to different mathematical principles, including the Pythagorean theorem and vector addition.
Step-by-step explanation:
To determine which congruence theorem to use to prove the congruence of two triangles, we need information about the sides and angles of the triangles. Here are the options:
- ASA Congruence Theorem: This requires two angles and the side between them to be congruent.
- AAS Congruence Theorem: This requires two angles and any side to be congruent.
- SSS Congruence Theorem: This requires all three sides to be congruent.
However, since the question does not provide specific information about the triangles in question, an accurate answer cannot be provided based on the data given. In typical scenarios, you would match the given information about the triangles to one of the theorems. For example, if you know two angles and the side between them are equal in both triangles, you'd use the ASA Congruence Theorem. Nevertheless, more information is needed to select the appropriate theorem in this case.
As for the other parts of the question which seem to refer to different mathematical concepts:
- The Pythagorean theorem is used in right triangles to relate the legs and the hypotenuse.
- The question about area comparisons (a. A₁ A₂ A3, b. A₁ = A₂ = A3, etc.) requires specific values or relationships between the areas to make a comparison.
- The Pythagorean theorem can indeed be used to calculate the magnitude of the resultant of two vectors that are perpendicular to each other, which is true.