Congruence Criteria

Understanding congruence criteria is essential in geometry, as it helps determine when two triangles are identical in shape and size. Key methods like SSS, SAS, ASA, and others provide reliable ways to prove triangle congruence.

1. Side-Side-Side (SSS) Congruence

   • If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
   • This criterion applies to all types of triangles, including scalene, isosceles, and equilateral.
   • SSS is a strong criterion because it does not depend on angles.

2. Side-Angle-Side (SAS) Congruence

   • 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 included angle is the angle formed between the two sides being compared.
   • SAS is useful for proving congruence when angle measures are known.

3. Angle-Side-Angle (ASA) Congruence

   • 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 included side is the side between the two angles being compared.
   • ASA is effective for proving congruence when angle measures are available.

4. Angle-Angle-Side (AAS) Congruence

   • 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.
   • AAS can be used when the side is not between the two angles.
   • This criterion is valid because the third angle is determined by the other two angles.

5. Hypotenuse-Leg (HL) Congruence for right triangles

   • 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.
   • HL is a specific case of SAS, applicable only to right triangles.
   • This criterion simplifies the process of proving congruence in right triangle scenarios.

6. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

   • Once two triangles are proven congruent, all corresponding sides and angles are also congruent.
   • CPCTC is often used as a justification in proofs after establishing triangle congruence.
   • This principle reinforces the relationship between triangle parts and overall congruence.

7. Reflexive Property of Congruence

   • Any geometric figure is congruent to itself.
   • This property is fundamental in proofs and helps establish congruence relationships.
   • It is often used to show that a side or angle in one triangle is equal to the corresponding side or angle in another triangle.

8. Symmetric Property of Congruence

   • 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 useful in proofs where the order of comparison may change.

9. Transitive Property of Congruence

   • 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 helps in establishing congruence across multiple figures.
   • It is often used in proofs to connect different congruence relationships.