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Congruent Triangles — Definition, Formula & Examples

Congruent triangles are triangles that have exactly the same shape and size, meaning all three pairs of corresponding sides are equal in length and all three pairs of corresponding angles are equal in measure. If two triangles are congruent, one can be placed exactly on top of the other through rotation, reflection, or translation.

Two triangles △ABC\triangle ABC and △DEF\triangle DEF are congruent, written △ABC≅△DEF\triangle ABC \cong \triangle DEF, if and only if there exists a one-to-one correspondence between their vertices such that AB=DEAB = DE, BC=EFBC = EF, AC=DFAC = DF, ∠A=∠D\angle A = \angle D, ∠B=∠E\angle B = \angle E, and ∠C=∠F\angle C = \angle F. The order of the vertices in the congruence statement specifies which parts correspond.

How It Works

You do not need to verify all six pairs of corresponding parts to prove two triangles are congruent. Instead, you can use one of five shortcut criteria: **SSS** (three pairs of sides equal), **SAS** (two sides and the included angle equal), **ASA** (two angles and the included side equal), **AAS** (two angles and a non-included side equal), or **HL** (hypotenuse and one leg equal, for right triangles only). To apply a criterion, identify which sides and angles you know are equal — often through given information, shared sides, vertical angles, or parallel-line angle relationships. Once you establish congruence, you can conclude that all remaining corresponding parts are also equal, a principle abbreviated as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Worked Example

Problem: In triangles ABC and DEF, you know that AB = DE = 7, BC = EF = 10, and the included angle ∠B = ∠E = 50°. Prove the triangles are congruent and find AC if DF = 8.
Identify known corresponding parts: We have two pairs of sides and the angle between them:
AB=DE=7,BC=EF=10,∠B=∠E=50°AB = DE = 7, \quad BC = EF = 10, \quad \angle B = \angle E = 50°
Choose the correct criterion: Two sides and the included angle match, so we use SAS (Side-Angle-Side).
State the congruence: By SAS, the triangles are congruent:
△ABC≅△DEF\triangle ABC \cong \triangle DEF
Apply CPCTC: Since corresponding parts of congruent triangles are congruent, AC must equal DF.
AC=DF=8AC = DF = 8
Answer: The triangles are congruent by SAS, and AC = 8.

Another Example

Problem: In triangle PQR and triangle XYZ, ∠P = ∠X = 40°, ∠Q = ∠Y = 75°, and PQ = XY = 12. Are the triangles congruent?
Check what is given: Two pairs of angles are equal, and the side between those angles (the included side) is equal:
∠P=∠X=40°,∠Q=∠Y=75°,PQ=XY=12\angle P = \angle X = 40°, \quad \angle Q = \angle Y = 75°, \quad PQ = XY = 12
Select the criterion: The equal side PQ lies between the two known angles in each triangle. This matches ASA (Angle-Side-Angle).
Conclude: By ASA:
△PQR≅△XYZ\triangle PQR \cong \triangle XYZ
Answer: Yes, the triangles are congruent by ASA.

Why It Matters

Triangle congruence is one of the most heavily tested topics in high school geometry and appears frequently on standardized tests like the SAT. Engineers and architects rely on congruence principles when designing structures with identical triangular components, ensuring each piece fits precisely. Mastering congruence proofs also builds the logical reasoning skills needed for later courses in mathematical proof and formal logic.

Common Mistakes

Mistake: Using SSA (two sides and a non-included angle) as a valid congruence criterion.
Correction: SSA does not guarantee congruence because two distinct triangles can share the same SSA measurements (the ambiguous case). Only SSS, SAS, ASA, AAS, and HL are valid.
Mistake: Writing the congruence statement with vertices in the wrong order, such as writing △ABC ≅ △FDE when the actual correspondence is A↔D, B↔E, C↔F.
Correction: The vertex order in the congruence statement defines which parts correspond. Always list vertices so that matching pairs align: △ABC ≅ △DEF means A↔D, B↔E, C↔F.

Related Terms