Triangle Congruence Proofs

A triangle congruence proof is a short argument with a predictable shape: start from the given facts, collect enough congruent pairs to satisfy a criterion (SSS, SAS, ASA, AAS, or HL), state that the triangles are congruent, and — if the problem asks about a specific side or angle — finish with CPCTC.

In the two-column format, every line has a statement on the left and a reason on the right. The statements are the facts you are claiming; the reasons are why you are allowed to claim them. Most proofs at this level are four to six lines long once you know the flow.

The shape of the proof

Step 1: state the givens, with the reason "Given." Step 2: add any congruence the diagram hands you for free — a shared side, vertical angles, a midpoint. Step 3: once you have three usable pairs, prove the triangles congruent by the matching criterion. Step 4: if the goal is a pair of corresponding parts, cite CPCTC.

Before writing anything, mark the diagram. Put tick marks and arcs on everything the givens tell you, then add the free congruences. When the marks show a complete SSS, SAS, ASA, AAS, or HL pattern, the proof writes itself.

Free congruences

The Reflexive Property: any segment or angle is congruent to itself. When two triangles share a side, that shared side is a congruent pair — reason: Reflexive Property.

Vertical angles: when two segments cross, the opposite angles at the intersection are congruent. Diagrams with an X-shaped crossing almost always use this.

Definitions also generate congruences. A midpoint splits a segment into two congruent segments; a bisector splits an angle into two congruent angles; parallel lines cut by a transversal give congruent alternate interior angles. Cite the definition or theorem by name.

In the crossing diagram below, $\overline{AC}$ and $\overline{BD}$ meet at $E$ . The tick marks show the given congruent segments, and the arcs mark the vertical angles $\angle AEB$ and $\angle CED$ — congruent for free — which sit between them to give SAS.

CPCTC comes last

CPCTC — corresponding parts of congruent triangles are congruent — is only valid after the triangle congruence is on the board. Using it earlier assumes the exact fact the proof is still building toward.

So the order is fixed: criterion first, CPCTC second. If a proof's goal is $\overline{BC} \cong \overline{EF}$ , the second-to-last line proves the triangles congruent and the last line states the goal with reason CPCTC.

Worked examples

Example 1: a shared side (SSS)

Given: $\overline{AB} \cong \overline{CB}$ and $\overline{AD} \cong \overline{CD}$ . Prove: $\triangle ABD \cong \triangle CBD$ .

State the first given

\overline{AB} \cong \overline{CB} \;\; \text{(Given)}

State the second given

\overline{AD} \cong \overline{CD} \;\; \text{(Given)}

The triangles share side

\overline{BD}

\overline{BD} \cong \overline{BD} \;\; \text{(Reflexive Property)}

Three pairs of congruent sides

\triangle ABD \cong \triangle CBD \;\; \text{(SSS)}

Answer: $\triangle ABD \cong \triangle CBD$ by SSS

Example 2: vertical angles (SAS)

Given: $\overline{AC}$ and $\overline{BD}$ intersect at $E$ , $\overline{AE} \cong \overline{CE}$ , and $\overline{BE} \cong \overline{DE}$ . Prove: $\triangle AEB \cong \triangle CED$ .

State the givens

\overline{AE} \cong \overline{CE},\; \overline{BE} \cong \overline{DE} \;\; \text{(Given)}

The crossing segments create congruent opposite angles

\angle AEB \cong \angle CED \;\; \text{(Vertical angles are congruent)}

The congruent angles sit between the two pairs of congruent sides

\triangle AEB \cong \triangle CED \;\; \text{(SAS)}

Answer: $\triangle AEB \cong \triangle CED$ by SAS

Example 3: finishing with CPCTC

You have proven $\triangle ABC \cong \triangle DEF$ by ASA. Justify the conclusion $\overline{BC} \cong \overline{EF}$ .

The triangle congruence is already established

\triangle ABC \cong \triangle DEF \;\; \text{(ASA)}

\overline{BC}

and

\overline{EF}

are corresponding sides — second and third letters of each name

\overline{BC} \leftrightarrow \overline{EF}

Corresponding parts of congruent triangles are congruent

\overline{BC} \cong \overline{EF} \;\; \text{(CPCTC)}

Answer: $\overline{BC} \cong \overline{EF}$ by CPCTC

Try one yourself

Common questions

What goes in each column of a two-column proof?

Statements on the left, reasons on the right. Every statement needs exactly one reason: Given, a definition, a property (like Reflexive), a theorem (like vertical angles), a congruence criterion, or CPCTC.

When do I use the Reflexive Property?

Whenever the two triangles share a side or an angle. The shared part is congruent to itself, and that line often supplies the third pair you need for SSS, SAS, or HL.

Does the order of the lines matter?

Yes. Each line may only rely on the givens and the lines above it. In particular, CPCTC must appear after the line that proves the triangles congruent — never before.

How do I decide which criterion to cite?

Count what you have marked: three sides is SSS; two sides with the included angle is SAS; two angles with the included side is ASA; two angles with a non-included side is AAS; and a right angle with congruent hypotenuses and legs is HL.

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Triangle Congruence Proofs

The shape of the proof

Free congruences

CPCTC comes last

Worked examples

Example 1: a shared side (SSS)

Example 2: vertical angles (SAS)

Example 3: finishing with CPCTC

Try one yourself

Common questions

Keep learning

Congruent Figures & Corresponding Parts (CPCTC)

SSS & SAS Triangle Congruence

ASA & AAS Triangle Congruence

HL Congruence for Right Triangles

Isosceles & Equilateral Triangle Theorems