Triangle congruence (SSS, SAS, ASA, AAS, HL) and proofs Knowledge-Based Study

Study notes

1. Definition of Congruence

   Two triangles are congruent when they have exactly the same size and shape. Formally, two triangles are congruent if and only if there is a correspondence between their vertices such that all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent.

   Congruence Statements

   When stating that two triangles are congruent, the order of vertices is critical because it identifies the corresponding parts. For example, the statement $\triangle ABC \cong \triangle DEF$ directly establishes the following:

   • Corresponding sides: $AB \cong DE$, $BC \cong EF$, and $AC \cong DF$
   • Corresponding angles: $\angle A \cong \angle D$, $\angle B \cong \angle E$, and $\angle C \cong \angle F$

   What is CPCTC?

   CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. In geometric proofs, CPCTC is often used as a justification immediately after proving that two triangles are congruent. Once congruence is established via an axiomatic shortcut, CPCTC allows you to deduce that any remaining pair of corresponding angles or sides is also congruent.

   Congruence Postulates and Theorems

   You do not need to measure all six pairs of parts (three sides and three angles) to prove two triangles are congruent. The following standard criteria guarantee congruence:

   • Side-Side-Side (SSS): If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
   • Side-Angle-Side (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. The included angle is the angle formed between the two given sides.
   • Angle-Side-Angle (ASA): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. The included side is the segment connecting the vertices of the two given angles.
   • Angle-Angle-Side (AAS): If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, the triangles are congruent.

   Hypotenuse-Leg (HL) Theorem

   The Hypotenuse-Leg (HL) Theorem applies exclusively to right triangles:

   If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.

   Why it works: By the Pythagorean theorem, knowing the lengths of the hypotenuse and one leg uniquely fixes the length of the remaining leg. Thus, HL is essentially a specialized application of SSS.

   Non-Valid Conditions: Why AAA and SSA Fail

   • Angle-Angle-Angle (AAA): Knowing three pairs of congruent angles guarantees that two triangles have the same shape, but not necessarily the same size. Therefore, AAA proves similarity, not congruence.
   • Side-Side-Angle (SSA): Two sides and a non-included angle do not uniquely determine a single triangle. In many cases, two distinct triangles can be formed with the same side lengths and angle measure (often known as the ambiguous case). Consequently, SSA cannot be used to prove congruence.

   Key Justifications in Triangle Proofs

   Geometric proofs rely on clear deductive reasoning. When proving triangle congruence, several geometric properties frequently provide the necessary congruences:

   • Reflexive Property of Congruence: A geometric figure is congruent to itself (for example, a shared segment $AB \cong AB$ or a shared angle $\angle A \cong \angle A$).
   • Vertical Angles Theorem: Vertical angles formed by intersecting lines are always congruent.
   • Definitions of Bisectors and Midpoints: A segment bisector divides a segment into two congruent segments; an angle bisector divides an angle into two congruent angles.
   • Parallel Line Theorems: If two parallel lines are cut by a transversal, alternate interior angles are congruent.

   Structure of a Two-Column Proof

   1. Identify the given statements and state what needs to be proven.
   2. Mark known congruent sides and angles on a mental or drawn diagram.
   3. Determine which valid criterion (SSS, SAS, ASA, AAS, or HL) connects the given information.
   4. List each step in a logical sequence with its corresponding mathematical reason or theorem.
   5. If required, use CPCTC in the final step to conclude that specific segments or angles are congruent.