Congruence of triangles

Two plane figures are congruent if they have exactly the same shape and size, so that placing one on top of the other covers it perfectly — no gaps, no overhang. For triangles this superposition test translates into a precise algebraic statement. $\triangle ABC$ is congruent to $\triangle DEF$, written

if there is a correspondence of vertices — here $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$ — under which every pair of corresponding sides is equal and every pair of corresponding angles is equal:

where: $AB, BC, CA$ are the sides of the first triangle and $DE, EF, FD$ the matching sides of the second, following the vertex correspondence $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$; $\angle A, \angle B, \angle C$ and $\angle D, \angle E, \angle F$ are the angles at those vertices. Six equalities in total — three sides, three angles — define congruence completely.

The order of letters is the whole claim. Writing $\triangle ABC \cong \triangle DEF$ asserts a specific matching: $A$ with $D$, $B$ with $E$, $C$ with $F$. The statement $\triangle ABC \cong \triangle EDF$ asserts a different matching ($A$ with $E$, $B$ with $D$) and is, for a generic pair of triangles, simply false even if the two triangles really are congruent under the correct correspondence. Always name congruent triangles so that matching vertices occupy the same position in both names — this single habit prevents most errors in the chapter.

CPCT — Corresponding Parts of Congruent Triangles are Equal. Once $\triangle ABC \cong \triangle DEF$ has been proved — usually with one of the congruence criteria SAS, ASA, SSS, or RHS, covered in the companion article on those criteria — you may conclude that every corresponding side and angle is equal, including pairs you never used in the proof. CPCT is not a new theorem to prove; it is simply what "congruent" already meant, applied to whichever side or angle you still need. Congruence must not be confused with equal area. Two triangles can enclose exactly the same area while having different shapes entirely — a tall thin triangle and a short wide one can match in area but never fit on top of one another. Congruence is the strictly stronger claim of identical shape and size.

Congruence has no single formula to derive — it is a definition, built directly from the idea of superposition — so the useful thing to nail down is why CPCT is valid, since that is the one logical step students misuse most.

Step 1. Start from what congruence actually asserts. $\triangle ABC \cong \triangle DEF$ is, by definition, the statement that all six corresponding measurements match: $AB = DE$, $BC = EF$, $CA = FD$, $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$. This is not something extra to prove after congruence — it is congruence.

Step 2. See why a criterion like SAS is a shortcut, not the definition. SAS lets you conclude all six equalities from just three of them (two sides and the included angle) because a triangle is rigid: fix two side lengths and the angle between them, and the third side and the other two angles are forced — there is only one triangle you can draw with those three measurements. So checking three matching parts is enough to guarantee all six.

Step 3. Read off the remaining three "for free." Once a criterion has verified three matching parts and rigidity guarantees the triangles are identical in shape and size, the other three parts — which you did not check — must also match, because they belong to the same, now fully determined, triangle. Naming this final step "CPCT" is a bookkeeping convenience: it flags that the equality being used comes from the definition of congruence itself, not from a fresh measurement or a fresh proof.

Step 4. Note the direction of logic. CPCT only runs from congruence to equal parts, never the reverse. You cannot assume two triangles are congruent because one pair of sides looks equal in a diagram — that is circular. Congruence must be established first (by a valid criterion), and only then does CPCT release the remaining equalities.

$\triangle PQR \cong \triangle XYZ$. Given $PQ = 5\text{ cm}$, $QR = 7\text{ cm}$, $\angle P = 40°$, and $\angle Q = 65°$, find $XY$, $YZ$, $\angle X$, $\angle Y$, and $\angle Z$.

Step 1. Read the correspondence directly from the congruence statement: $\triangle PQR \cong \triangle XYZ$ means $P \leftrightarrow X$, $Q \leftrightarrow Y$, $R \leftrightarrow Z$, in that order.

Step 2. Apply CPCT to the sides and angles that are directly matched: $XY = PQ = 5\text{ cm}$ and $YZ = QR = 7\text{ cm}$; likewise $\angle X = \angle P = 40°$ and $\angle Y = \angle Q = 65°$.

Step 3. Find $\angle R$ first, since it was not given directly, using the angle sum property of a triangle: $\angle P + \angle Q + \angle R = 180°$, so $40° + 65° + \angle R = 180°$, giving $\angle R = 75°$.

Step 4. Apply CPCT one more time: since $\angle Z = \angle R$ under the correspondence, $\angle Z = 75°$.

Answer: $XY = 5\text{ cm}$, $YZ = 7\text{ cm}$, $\angle X = 40°$, $\angle Y = 65°$, $\angle Z = 75°$.

Competitive stretch

The same figure often hides a second question: given $\triangle ABD \cong \triangle ACD$ where $AB = AC$ and $AD$ bisects $\angle BAC$ with $D$ on $BC$, CPCT gives $BD = CD$, $\angle ABD = \angle ACD$, and $\angle ADB = \angle ADC$. Since $B$, $D$, $C$ are collinear, $\angle ADB + \angle ADC = 180°$; combined with $\angle ADB = \angle ADC$ from CPCT, this forces $2\angle ADB = 180°$, so $\angle ADB = \angle ADC = 90°$ — the bisector of the apex angle of an isosceles triangle is perpendicular to the base, a fact that resurfaces later in properties of triangles.

• Writing the congruence statement with vertices in the wrong order — since the order is the correspondence, $\triangle ABC \cong \triangle DEF$ and $\triangle ABC \cong \triangle EDF$ are different claims, and only one can be true.
• Assuming two triangles are congruent just because they "look the same size" in a diagram, with no stated equal sides or angles and no proof via a congruence criterion.
• Confusing congruence (same shape and size) with equal area or with similarity (same shape, possibly different size) — equal area never implies congruence, and Class 9 does not cover similarity at all.
• Using CPCT before congruence has actually been proved — CPCT is a conclusion you draw only after establishing $\triangle ABC \cong \triangle DEF$ with a valid criterion, never a starting assumption.

Ask the tutor: If you're unsure how to read the correct vertex correspondence off a diagram, or want to check whether a CPCT step in your own proof is valid, ask the AI tutor to walk through it with you step by step.