Congruence Criteria for Triangles to Know for Elementary Algebraic Geometry

Understanding triangle congruence is key in geometry. The criteria—SSS, SAS, ASA, AAS, and HL—help determine when triangles are identical in shape and size. These principles connect to broader concepts in Elementary Algebraic Geometry, enhancing our grasp of geometric relationships.

1. Side-Side-Side (SSS) Congruence Criterion

   • States that if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
   • All corresponding sides must be measured and compared.
   • This criterion applies to all types of triangles, including scalene, isosceles, and equilateral.

2. Side-Angle-Side (SAS) Congruence Criterion

   • States that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
   • The angle must be between the two sides being compared.
   • This criterion is useful for proving congruence without needing to measure all three sides.

3. Angle-Side-Angle (ASA) Congruence Criterion

   • States that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
   • The side must be between the two angles being compared.
   • This criterion emphasizes the importance of angle measures in determining triangle congruence.

4. Angle-Angle-Side (AAS) Congruence Criterion

   • States that if two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent.
   • The non-included side can be any side, not necessarily between the two angles.
   • This criterion is particularly useful when angle measures are known.

5. Hypotenuse-Leg (HL) Congruence Criterion for Right Triangles

   • States that if the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, the triangles are congruent.
   • This criterion is specific to right triangles and simplifies the process of proving congruence.
   • It is a special case of the SAS criterion.

6. Reflexive Property of Congruence

   • States that any geometric figure is congruent to itself.
   • This property is fundamental in proofs and helps establish relationships between figures.
   • It is often used to show that a side or angle is common to two triangles.

7. Symmetric Property of Congruence

   • States that if one figure is congruent to another, then the second figure is congruent to the first.
   • This property allows for flexibility in reasoning about congruence.
   • It is essential for establishing equivalence in geometric proofs.

8. Transitive Property of Congruence

   • States that if one figure is congruent to a second figure, and the second figure is congruent to a third figure, then the first and third figures are congruent.
   • This property is crucial for chaining congruences together in proofs.
   • It helps in establishing relationships among multiple triangles or geometric figures.