Congruent Triangle Calculator

The Congruent Triangle Calculator checks whether two triangles are congruent (identical in size and shape) using the five accepted congruence postulates: SSS, SAS, ASA, AAS, and HL. This tutorial walks through how to recognize which postulate applies in any given problem, the worked examples for each, and the common "traps" (SSA and AAA) that look like they should work but don't.

What congruence means precisely

Two triangles are congruent (written △ABC ≅ △DEF) when ALL of the following hold:

• Side AB = side DE (in length)
• Side BC = side EF
• Side CA = side FD
• Angle A = angle D
• Angle B = angle E
• Angle C = angle F

That's 6 equalities. But you don't need to verify all 6 — the postulates below let you conclude congruence from a smaller, sufficient set.

Quick decision tree — which postulate to use

SSS — the simplest case

If all three sides of one triangle equal the three sides of another, the triangles are congruent. Order matters when matching: pair the longest side with the longest in the other triangle, the middle with middle, the shortest with shortest.

Worked example: Triangle 1 has sides 5, 7, 9. Triangle 2 has sides 9, 7, 5. The triangles ARE congruent (SSS), even though the sides are listed in different orders.

SAS — two sides + included angle

If two sides and the angle BETWEEN them are equal, the triangles are congruent. The angle must be sandwiched between the two given sides.

Worked example: Triangle 1 has sides AB = 6 and AC = 8 with included ∠A = 50°. Triangle 2 has sides DE = 6 and DF = 8 with ∠D = 50°. SAS → congruent.

What does NOT work: Two sides AB = 6, AC = 8, and angle ∠B = 50° — that's SSA (angle is NOT between the given sides). SSA is the famous "ambiguous case" — it can produce zero, one, or two triangles.

ASA — two angles + included side

If two angles and the side BETWEEN them are equal, the triangles are congruent. Once two angles are known, the third is forced (sum = 180°), and the side fixes the scale.

Worked example: Triangle 1 has ∠A = 40°, side AB = 7, ∠B = 80°. Triangle 2 has ∠D = 40°, side DE = 7, ∠E = 80°. ASA → congruent.

ASA appears most often in proofs involving parallel lines, because parallel-line theorems give you angle equalities "for free" and you often have one shared/given side.

AAS — two angles + non-included side

The fourth postulate is essentially ASA with the side moved. Two angles equal + one side equal (the side does NOT sit between the two given angles) → triangles congruent.

Worked example: Triangle 1 has ∠A = 35°, ∠B = 75°, side BC = 9 (opposite to ∠A, not between A and B). Triangle 2 has ∠D = 35°, ∠E = 75°, side EF = 9. AAS → congruent.

Why AAS works: knowing two angles forces the third, which makes the case equivalent to ASA after determining the third angle.

HL — the right-triangle-only postulate

For right triangles, knowing the hypotenuse and one leg is enough. This works because a right triangle's 90° angle, when added to a known leg and hypotenuse, forces the third side via the Pythagorean theorem.

Worked example: Right triangle 1 has hypotenuse 13, one leg 5 (other leg by Pythagoras = 12). Right triangle 2 has hypotenuse 13, leg 5 (other leg = 12). HL → congruent.

HL is essentially SSA with a right angle — SSA doesn't work in general, but the 90° angle eliminates the ambiguous case. See the Right Triangle Congruence Calculator for the specialized version with HA, LA, and LL.

The traps — SSA and AAA

Two letter combinations look like they should work but don't:

SSA (Side-Side-Angle, non-included): ambiguous. The same SSA inputs can fit zero, one, or two triangles. Why: the unknown side can "swing" to two positions, both valid. Exception: SSA with a right angle (= HL) works because the right angle removes the ambiguity.

AAA (Angle-Angle-Angle): equal angles only prove SIMILARITY, not congruence. A small triangle and a huge dilation of it have the same angles but different sizes. AAA gives "same shape", not "same shape AND size".

ASS = SSA: just SSA spelled backwards. Same ambiguity.

The 6-step decision tree (in practice)

1. Is the triangle right and given hypotenuse + leg? → HL.
2. All 3 sides given? → SSS.
3. 2 sides + included angle? → SAS.
4. 2 angles + included side? → ASA.
5. 2 angles + non-included side? → AAS.
6. Anything else (just SSA without a right angle, just AAA)? → NOT sufficient — need more info.

CPCTC — the universal post-step

Once you've proven two triangles congruent by ANY postulate, you can conclude that ALL six pairs of corresponding parts (3 sides + 3 angles) are equal. This is CPCTC: Corresponding Parts of Congruent Triangles are Congruent.

CPCTC is the standard final step of proofs that conclude "two segments are equal" or "two angles are equal" by going through the intermediate step of proving the containing triangles congruent.

Common mistakes

• Using SSA as if it were a postulate. SSA is the ambiguous case — it can produce zero, one, or two triangles. NOT sufficient for congruence.
• Treating AAA as congruence. Equal angles prove similarity (same shape). Adding "and same scale" requires a side too. AAA alone proves only similarity.
• Matching the wrong sides in SSS. Make sure you pair longest with longest, middle with middle, shortest with shortest. Crossing these pairs gives a false congruence.
• Forgetting "included" vs "non-included". SAS requires the angle BETWEEN the two given sides. ASA requires the side BETWEEN the two given angles. Get this wrong and the postulate doesn't apply.
• Citing the postulate without showing the elements. A complete proof must explicitly state which two sides + which angle (or which two angles + which side). "Therefore congruent by SAS" alone is incomplete.