SSS, SAS, ASA, and AAS Theorems

Learning Objectives Identify the necessary conditions to prove triangle congruence using SSS, SAS, ASA, and AAS. Apply the SSS, SAS, ASA, and AAS theorems to determine if two triangles are congruent. Write a logical, step-by-step geometric proof to demonstrate triangle congruence. Distinguish between valid congruence theorems and invalid shortcuts like SSA and AAA. Correctly write a triangle congruence statement, ensuring corresponding vertices are in the correct order. Identify what additional information is needed to prove two triangles are congruent. Use the concept of 'Corresponding Parts of Congruent Triangles are Congruent' (CPCTC) after proving congruence. How can a manufacturer produce thousands of identical triangular parts for a machine without measurin...

TermDefinitionExample Congruent TrianglesTwo triangles are congruent if all three corresponding sides and all three corresponding angles are equal. Congruent triangles are exact copies of each other in terms of size and shape.If ΔABC ≅ ΔDEF, it means AB = DE, BC = EF, AC = DF, ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F. Corresponding PartsThe sides and angles that are in the same relative position in two different figures. In a congruence statement like ΔABC ≅ ΔDEF, ∠A corresponds to ∠D, and side BC corresponds to side EF.In ΔPQR ≅ ΔXYZ, the side corresponding to PR is XZ. Included AngleThe angle formed between two specified sides of a triangle.In ΔABC, ∠B is the included angle between sides AB and BC. Included SideThe side located between two specified angles of a triangle.In ΔABC, side AC is the inc...

SSS (Side-Side-Side) Congruence Theorem If three sides of one triangle are congruent to the three corresponding sides of another triangle, then the two triangles are congruent. Use this when you have information about all three pairs of corresponding sides and no information about angles. SAS (Side-Angle-Side) Congruence Theorem If two sides and the *included* angle of one triangle are congruent to the two corresponding sides and *included* angle of another triangle, then the two triangles are congruent. Use this when you know two pairs of sides are congruent and the angle *between* those sides is also congruent. ASA (Angle-Side-Angle) Congruence Theorem If two angles and the *included* side of one triangle are congruent to the two corresponding angles and *included* s...