Final answer:
Triangles are congruent when they have the same shape and size, meaning all corresponding sides and angles match. Congruency is determined based on geometric postulates and can be verified using criteria like SSS, SAS, ASA, and AAS. Trigonometry depends on congruent triangle relationships through consistent trigonometric ratios.
Step-by-step explanation:
Understanding Congruent Triangles
Triangles are considered congruent when they have the same shape and size. This implies that all corresponding sides are equal in length, and all corresponding angles are equal in measure. When two triangles are congruent, you can place one over the other, and they will match perfectly. To determine if triangles are congruent, mathematicians use specific criteria such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS).
To explain why triangles are congruent, one must understand that congruence is a product of corresponding elements being identical. This is based on geometric postulates that serve as the foundational rules in geometry. For instance, in the context of the Pythagorean Theorem, if we apply it to two identical right triangles, we'll find that the results are the same because the triangles are congruent by definition, their parts correspond, and they follow the same logical postulates.
Trigonometry is the branch of mathematics dealing with the relationships between the angles and sides of triangles. The trigonometric ratios, such as sine, cosine, and tangent, are based on the angles of right triangles and are consistent due to the congruent nature of triangles that are identical in their angle measures and side lengths.