What geometric principle is applied to prove triangles need to fulfill the criteria of their congruency?

The Congruence Criteria Theorem is fundamental in determining if two triangles are congruent. This theorem outlines specific sets of conditions that, when met, establish the equality of the angles and sides of two triangles. The primary criteria include Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) for right triangles.

For instance, if two triangles meet the criteria of SSS, where all three sides of one triangle are equal in length to the three sides of another triangle, it can be concluded that the two triangles are congruent. This congruence indicates that not only are the sides equal, but the angles are also equal, thus proving the two triangles are identical in shape and size.

Other principles mentioned, like the Pythagorean Theorem, focus on relationships involving right triangles and their sides, while the Isosceles Triangle Theorem relates specifically to the properties of isosceles triangles. The Midpoint Theorem deals with line segments and does not directly address triangle congruence. Therefore, the Congruence Criteria Theorem is the appropriate principle that