Triangle Congruence Calculator - SSS, SAS, ASA, AAS, and HL Tests

The triangle congruence calculator tests whether two triangles are congruent by applying one of the five shortcut rules (SSS, SAS, ASA, AAS, or HL) and comparing the matching parts inside a small tolerance. Pick a rule, enter the three matching parts for each triangle, and the calculator returns a Congruent or Not Congruent verdict along with a per-part difference so you can see which part did not line up.

• • Geometry homework and proofs: Confirm the SSS, SAS, ASA, or AAS argument in a two-column proof.
• • Quick congruence checks: Skip the full six-part comparison by entering the three matching parts that your chosen rule lists as enough.
• • Right triangle shortcut (HL): Confirm two right triangles are congruent from the hypotenuse and one leg.
• • Engineering and CAD tolerance checks: Set a small tolerance to decide whether two triangulated parts land inside the allowed build tolerance.

Two triangles are congruent when one can be placed exactly on top of the other by a rigid motion so that every side and every angle matches. Congruence is stronger than similarity because similar triangles can differ by a scale factor, while congruent triangles must share the same size and shape. The five shortcut rules each list the minimum set of matching parts that is enough to prove congruence. Keep all three parts in the same unit and enter them in the same order; the calculator sorts the parts internally for SSS and HL, but for SAS, ASA, and AAS the included or non-included part must sit in the slot the rule requires.

For shape-only comparisons that allow a scale factor, the Similar Triangles Calculator tests the proportion a / b = c / d and reports the scale factor and area ratio instead of a yes/no verdict.

The triangle congruence calculator reads the chosen rule, validates the six inputs, sorts the parts where the rule allows it, computes the absolute difference for each of the three matching part pairs, and counts how many pairs fall inside the tolerance. If all three match, the verdict is Congruent.

• criterion: Congruence rule (SSS, SAS, ASA, AAS, or HL).
• a1, b1, c1: Three matching parts of triangle 1. Sides in length units; angles in degrees.
• a2, b2, c2: Three matching parts of triangle 2 in the same order and unit.
• tolerance: Smallest absolute difference treated as equal. Default 0.01.

For SSS and HL the calculator sorts the three side parts of each triangle before comparing, so the labeling of side a, b, and c does not matter. For SAS, ASA, and AAS the parts are already in correspondence; enter the included part in the c slot. A tolerance of 0.01 on a side length treats 4.005 and 4.011 as equal; 0.5 on an angle treats 60.2 and 60.7 degrees as equal.

According to Wolfram MathWorld, two figures exhibit geometric congruence when one can be transformed into the other by an isometry, and the SSS, SAS, ASA, AAS, and HL criteria each give a minimum set of matching parts that prove two triangles congruent.

When the three sides are given, the SSS Triangle Calculator solves the same Side-Side-Side case from the given sides to a full triangle.

These four ideas decide which rule applies, how strict the comparison is, and how the per-part tolerance is interpreted.

The four general rules work on any triangle, while HL only works on right triangles. SSA and AAA are deliberately not rules: SSA can produce two different triangles with the same parts, and AAA only proves similarity. Choosing the rule that matches the given parts is the first step of any congruence proof: three sides pick SSS, two sides and the included angle pick SAS, two angles and the included side pick ASA, two angles and a non-included side pick AAS, and right triangles with hypotenuse and one leg pick HL.

HL sits on top of the Pythagorean theorem, so the Pythagorean Triples Calculator lists the integer triples (3-4-5, 5-12-13, 8-15-17) that satisfy a squared plus b squared equals c squared.

Pick a rule, enter the three matching parts for each triangle, set the tolerance, and read the verdict plus the per-part difference.

1. 1 Choose the congruence rule: Set the criterion dropdown to SSS, SAS, ASA, AAS, or HL based on the parts the problem gives you.
2. 2 Enter the three parts of triangle 1: Type the three matching parts in slots a, b, and c. Sides go in length units and angles go in degrees.
3. 3 Enter the three parts of triangle 2 in the same order: For SSS and HL the labeling does not matter (the calculator sorts); for SAS, ASA, and AAS the included part must sit in the c slot.
4. 4 Set the tolerance: Keep 0.01 for exact side measurements. Bump to 0.1 or 0.5 for angles or measured values with rounding noise.
5. 5 Read the verdict and the per-part difference: The primary result is Congruent or Not Congruent. The three deltas show the absolute difference for part a, b, and c.

A geometry student has two triangles with sides 6, 8, 10 and 10, 6, 8. They set the criterion to SSS, enter the sides in the a, b, c slots, leave the tolerance at 0.01, and the calculator reports Congruent because the sorted sides (6, 8, 10) match in both triangles.

For right triangles in particular, the Right Triangle Calculator extends the hypotenuse-and-leg pair from HL into a full triangle solution.

Putting all five congruence rules in one calculator keeps the workflow short and avoids looking up a separate tool for each rule.

• • Five rules in one place: SSS, SAS, ASA, AAS, and HL are all available from the same criterion picker.
• • Sorts sides for SSS and HL: The calculator sorts the three side parts before comparing.
• • Per-part difference makes the failure visible: Delta a, delta b, and delta c are returned next to the verdict.
• • Tolerance handles measurement noise: A configurable tolerance lets the same calculator handle exact and measured values.
• • Honest input validation: Negative sides, out-of-range angles, and bad side triples are caught with a clear error.

The biggest practical win is the per-part difference. A two-column proof can show that two sides and an included angle are equal, but the calculator prints the actual delta so a grader or study partner can see how close the match was. If a design has to be congruent to a reference within a 0.05 mm build tolerance, the calculator tells the user whether the part is in or out of spec.

Once the congruence test passes, the Triangle Calculator extends the same side inputs to angles, perimeter, inradius, and area.

The verdict is exact within the chosen tolerance, but the inputs and the chosen rule decide how much weight the result carries.

• • The calculator compares the three parts you enter for the chosen rule and nothing more.
• • HL is for right triangles only. If the longest side is not actually the hypotenuse, the rule may give a false Congruent.
• • A Not Congruent result does not say which other rule might have worked, so if the chosen rule fails, try a different rule on the same triangle pair.

If the verdict is Not Congruent and you expected Congruent, the first thing to check is the labeling: did the included or non-included part land in the c slot for the rule you chose? The second thing is the tolerance: 0.01 is too tight for protractor readings, so raising the tolerance to 0.5 often turns a false Not Congruent into a real Congruent.

Once the rule that the parts describe is identified, the AAS Triangle Calculator can take a known AAS pair straight through to a fully solved triangle, which is useful when a Not Congruent verdict on a mixed pair points the user toward a different rule.

According to Khan Academy, the SSS, SAS, ASA, and AAS postulates each give a minimum set of parts whose equality is enough to declare two triangles congruent without measuring all six parts.

According to Math Open Reference, SSS, SAS, ASA, and AAS are the four triangle congruence rules, and HL is a right-triangle-only rule that uses the hypotenuse and one leg.