Triangle Congruence Calculator

Triangle congruence relies on proving that two triangles are exact copies of each other, meaning all three corresponding sides and three corresponding angles are equal. However, you do not need all six measurements to prove congruence. Standard geometric theorems allow us to verify congruence using subsets of information: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS). When missing values occur, the calculator applies fundamental laws like the Law of Cosines (c² = a² + b² - 2ab cos(C)) and the Law of Sines (a / sin(A) = b / sin(B) = c / sin(C)) to solve for unknown sides or angles before performing the final congruence check.

Consider testing two triangles to see if they are congruent. Suppose Triangle 1 is defined by the Side-Angle-Side (SAS) criterion where side a = 5, side b = 7, and the included angle C = 60 degrees (pi/3 radians). Triangle 2 is defined with side a = 5, side b = 7, and included angle C = 60 degrees. First, the calculator determines the third side (c) of Triangle 1 using the Law of Cosines: c = sqrt(5² + 7² - 2(5)(7)cos(60°)) = sqrt(25 + 49 - 70(0.5)) = sqrt(74 - 35) = sqrt(39) ≈ 6.24. It repeats this calculation for Triangle 2. Because all corresponding sides (5, 7, and 6.24) and the included angle match identically, the algorithm verifies that Triangle 1 and Triangle 2 are congruent under the SAS theorem.

When inputting angle measurements, ensure your calculator mode matches your data format, whether using degrees or radians. Always pay close attention to the position of the angle; the Angle-Side-Side (commonly known as SSA) configuration can lead to ambiguous cases or non-congruent shapes unless the angle is strictly between two known sides (making it SAS). Double-check your known inputs to ensure they correspond to the same relative vertices on both shapes to prevent false mismatch results.