Triangle Congruence Explorer

Enter measurements for two triangles, pick a congruence or similarity criterion, and instantly see whether they match. SVG diagrams overlay both triangles so you can visually confirm the result.

Explore by Criterion

Select a congruence or similarity criterion, fill in the required measurements for both triangles, then check the result.

Congruence Criteria

Similarity Criteria

SSS — Side-Side-Side: All three sides equal

Triangle 1

Triangle 2

Key Formulas

Law of Cosines:
Law of Sines:

Reference Guide

Congruence Criteria

Two triangles are congruent when they have identical shape and size. Five criteria guarantee congruence:

  • SSS - all three pairs of sides are equal.
  • SAS - two sides and the angle between them are equal.
  • ASA - two angles and the side between them are equal.
  • AAS - two angles and a non-included side are equal.
  • HL - right triangles with equal hypotenuse and one leg.

Note: SSA (two sides and a non-included angle) does NOT guarantee congruence due to the ambiguous case.

Similarity Criteria

Two triangles are similar when they have the same shape but not necessarily the same size. Three criteria guarantee similarity:

  • AA - two pairs of angles are equal. Since angles sum to 180 degrees, two equal angles force the third to match too.
  • SSS~ - all three pairs of sides are in the same ratio.
  • SAS~ - two pairs of sides are in the same ratio and the included angles are equal.

If two triangles are congruent they are also similar with scale factor 1.

Law of Cosines

Used when you know three sides (SSS) or two sides and an included angle (SAS).

General form
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
Solving for angle C (SSS)
C=cos⁡−1 ⁣(a2+b2−c22ab)C = \cos^{-1}\!\left(\frac{a^2 + b^2 - c^2}{2ab}\right)

Law of Sines

Used when two angles and any side are known (ASA or AAS).

General form
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

For ASA, first compute the third angle (C=180∘−A−BC = 180^\circ - A - B), then use the Law of Sines to find the remaining sides.

Standard Triangle Notation

Vertices and Sides

Vertices are labeled A, B, C. Side aa is opposite vertex A, side bb is opposite vertex B, and side cc is opposite vertex C.

Angles in Degrees

Angles A, B, C are measured in degrees and must all be positive with A+B+C=180∘A + B + C = 180^\circ.

Right Triangle (HL)

For HL, the right angle is always at C so C=90∘C = 90^\circ. Side cc is the hypotenuse and sides aa, bb are the legs.