Final answer:
Triangles are congruent when they have the same size and shape, with corresponding sides and angles being equal. To prove congruence, one can use postulates like SSS, SAS, ASA, or AAS, and employ trigonometry and the Pythagorean Theorem for verification in right triangles.
Step-by-step explanation:
Triangles can be congruent when they have the exact same size and shape. This means each corresponding side and angle must be equal. When thinking about a triangle, we envision a three-sided figure lying on a plane, with the sum of its angles totaling 180 degrees. One method to prove triangles are congruent involves using postulates such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), or Angle-Angle-Side (AAS).
For example, if we know that two triangles each have two sides and the angle between those sides (SAS) that are identical in length and measure, then the triangles are congruent. Similarly, if two triangles have three pairs of sides that are respectively equal in length (SSS), they are also congruent. Trigonometry can be used to verify congruence when given angles and sides, as calculations using trigonometric functions should yield the same results as using the Pythagorean Theorem for right triangles.