3.1 Triangle Congruence Criteria

Why Congruence Criteria Matter

Two triangles are congruent when one can be mapped exactly onto the other by a sequence of rigid motions (translations, reflections, and rotations). Congruent triangles have all three pairs of corresponding sides equal and all three pairs of corresponding angles equal, which is six equalities in total. On the Geometry Regents you almost never verify all six. Instead you use a congruence criterion, a minimal shortcut that guarantees congruence from just three pieces of information. Congruence is the single largest domain on the exam (about 27-34% of credit), and nearly every Part II-IV proof opens by naming one of these criteria.

The Five Valid Criteria (NY Next Gen GEO-G.CO.8)

SSS (Side-Side-Side) works because a triangle with three fixed side lengths is rigid, so there is only one such triangle. SAS (Side-Angle-Side) requires the marked angle to sit between the two marked sides. ASA (Angle-Side-Angle) requires the marked side to lie between the two marked angles. AAS (Angle-Angle-Side) uses two angles and a side that is not between them; it is valid because the third angle is forced by the 180-degree angle sum, which quietly converts AAS back into ASA.

HL: The Right-Triangle Special Case

HL (Hypotenuse-Leg) applies only when both triangles are already known to be right triangles. If the two hypotenuses are congruent and one pair of legs is congruent, the triangles are congruent. HL is essentially SSA in disguise, but the built-in right angle removes the ambiguity, so it is valid, and only valid, for right triangles. A frequent trap is invoking HL before a right angle has actually been established.

Why SSA and AAA Fail

Two tempting combinations do not prove congruence:

• SSA / ASS uses two sides and a non-included angle. This is the ambiguous case: the side opposite the given angle can swing to two different positions, producing two triangles that are not congruent. SSA is only safe when that angle is a right angle, which is precisely the HL situation.
• AAA uses three pairs of angles. Equal angles lock in the same shape but not the same size, so AAA guarantees the triangles are similar, never necessarily congruent. The exam tests this congruent-versus-similar distinction directly.

Memory hook: a valid criterion always includes at least one side and never rests on angle information alone.

Correspondence: Order Tells the Story

When you write triangle ABC is congruent to triangle DEF, the letter order encodes the correspondence: A matches D, B matches E, C matches F, and therefore side AB matches DE, BC matches EF, and AC matches DF. Getting this order right is essential because the very next proof step, CPCTC, reads corresponding parts straight off the congruence statement. A Part I multiple-choice item may simply ask which angle corresponds to angle B, and the answer (angle E) comes directly from the statement order, not from the picture.

Reading the Marks in a Diagram

Regents figures communicate the given information with tick marks and arc marks. Matching tick marks mean segments are congruent; matching arc marks mean angles are congruent; a small square means a right angle. Before choosing a criterion, translate the picture into a list:

1. Mark every congruent side and angle the problem explicitly gives you.
2. Add the free congruences the figure hands you: a shared side is congruent to itself (the Reflexive Property), and vertical angles formed by two crossing lines are congruent.
3. Count how many sides and angles you now have, and check their positions (included versus non-included).

Worked Example: Choosing the Criterion

In triangles ABC and ADC, you are told AB is congruent to AD and angle BAC is congruent to angle DAC, and the two triangles share side AC. You have AB congruent to AD (one pair of sides), angle BAC congruent to angle DAC (one pair of angles, sitting between the marked sides), and AC congruent to AC by the Reflexive Property (a second pair of sides). That is Side-Angle-Side with the angle included between the two sides, so the criterion is SAS. The shared side did the quiet work, which is typical: spotting the reflexive side is often the whole puzzle.

Common Traps

• Using HL without first stating the triangles are right triangles.
• Treating SSA as valid (it is the ambiguous case).
• Confusing included with non-included parts (SAS needs the angle inside the two sides; ASA needs the side inside the two angles).
• Choosing AAA and concluding congruence instead of similarity.
• Forgetting the free reflexive side or vertical angles the diagram provides.