3.2 Writing Triangle Congruence Proofs & CPCTC

From Diagram to Two-Column Proof

Parts II-IV of the Geometry Regents reward justification, not just a correct final answer. The workhorse format is the two-column proof: a numbered list of statements on the left, each paired with a reason on the right. Every statement must be a given fact, a definition, a postulate, or a previously proven theorem. The arc of a triangle-congruence proof is always the same: (1) collect congruent parts, (2) invoke a congruence criterion (SSS, SAS, ASA, AAS, or HL), and, when the goal is a segment or angle inside the triangles, (3) finish with CPCTC.

The Reasons Toolbox

Most triangle proofs draw their reasons from a short, reusable list. Memorize it so the right justification is always at hand:

CPCTC: The Follow-Through

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. It lets you conclude that a specific pair of sides or angles is equal, but only after the triangles have been proven congruent. Order is absolute: CPCTC can never justify the triangle congruence itself; it is always the step immediately after SSS, SAS, ASA, AAS, or HL. Whenever a proof asks you to show two segments are equal, that a segment bisects an angle, or that two lines are parallel, the plan is almost always the same phrase: prove the triangles congruent, then apply CPCTC.

A Full Worked Proof

Given: M is the midpoint of BD, and AB is parallel to ED. Prove: AB is congruent to ED.

Strategy: AB and ED are corresponding sides of triangles ABM and EDM. Prove those two triangles congruent, then release the sides with CPCTC.

Trace the logic: line 2 supplies a side, line 4 supplies an angle, and line 5 supplies a second angle. The side BM sits between angles ABM and AMB, so the pattern is Angle-Side-Angle, giving ASA in line 6. Only once congruence is established can line 7 pull out the corresponding sides with CPCTC. Reverse the order and the proof collapses.

Building the Plan Before You Write

Before writing a single line, mark the diagram: transfer every given onto the figure, then add the free congruences (reflexive sides, vertical angles). Ask which two triangles contain the parts you must prove equal, and whether you now hold SSS, SAS, ASA, AAS, or HL. Choosing the criterion first tells you exactly which three congruent parts to establish, so the statement column almost writes itself.

Why Two Columns

NYSED accepts a logically valid proof in two-column, paragraph, or flow-chart form, but the two-column layout is safest under time pressure because it forces a reason beside every statement and makes any gap obvious to a rater. Whatever format you pick, the mathematical content, the ordered chain from givens to conclusion, must be complete to earn full credit.

A Second Pattern: Proving a Bisector

Many proofs end not with a segment but with a claim such as "ray AD bisects angle BAC" or "C is the midpoint of AB." The plan is identical: prove two triangles congruent, use CPCTC to obtain the equal parts, then invoke a definition. For a bisected angle, CPCTC gives angle BAD congruent to angle CAD, and the definition of an angle bisector finishes the proof. For a bisected segment, CPCTC gives the two halves congruent, and the definition of a midpoint closes it. Recognizing that the final line is a definition, not CPCTC itself, is a common place students drop the last credit.

Common Proof-Writing Traps

• Skipping reasons. Every statement needs a justification; a bare statement earns no credit.
• Using CPCTC too early. It follows the congruence line and never precedes it.
• Naming the wrong criterion. Confirm the side is included (SAS, ASA) or non-included (AAS) exactly as claimed.
• Mis-ordering the congruence statement. The corresponding vertices must be listed in order so CPCTC reads correctly.
• Assuming from the picture. A segment that merely looks bisected is not bisected unless it is given or proven.