Math solver triangle congruence: check SSS, SAS, ASA, AAS, HL

Triangle congruence calculator: what to enter

Two triangles are congruent when one fits exactly on the other, turned over if needed. Three well-chosen parts are enough to know this, and the calculator checks which of the five standard tests your numbers satisfy.

• Sides go in the first row of each triangle: side a is opposite corner A, b opposite B, c opposite C (d, e, f for triangle 2).
• Angles go in the second row, in degrees. Give at least three measures per triangle in total. Two angles give the third.
• Values are compared with a small allowance for rounding (sides within 0.2%, angles within 0.1°), so 53.13° matches 53.1301°, but 5 does not equal 5.5.

How to use the math solver triangle congruence tool

1. Fill in the sides and angles you know for triangle 1 (corners A, B, C) and triangle 2 (corners D, E, F). Side a is opposite corner A, side b opposite B, and so on. Leave unknown boxes empty.
2. Press Check congruence, or try one of the example buttons.
3. Read the verdict first: congruent with the test named, not congruent, or cannot tell.
4. Read the steps to see which parts were matched and which were worked out, and the corner pairing for your congruence statement.

Impossible input is reported, not guessed at: sides 3, 4 and 9 cannot form a triangle, because 3 + 4 is less than 9.

SSS SAS ASA AAS HL: the five congruence tests

Each test names three matching parts that fix a triangle completely.

• SSS (side, side, side). All three sides of one triangle equal the three sides of the other. No angles needed.
• SAS (side, angle, side). Two sides and the angle between them. The angle must sit between the two sides, at the corner where they meet.
• ASA (angle, side, angle). Two angles and the side between them, which is the side that joins the two corners.
• AAS (angle, angle, side). Two angles and a side that is not between them. It follows from ASA, because once two angles match, the third matches too, and then any side lies between a pair of known angles.
• HL (hypotenuse, leg). Only for right triangles: the hypotenuses are equal and one pair of legs is equal. The hypotenuse is the side opposite the right angle.

When a test applies, every other part is equal too (CPCTC, corresponding parts of congruent triangles). To compare triangles described in different ways, the calculator first works out the missing parts of each with the law of cosines and the law of sines.

Triangle congruence solver: worked examples

SAS. Triangle 1 has b = 7, c = 9 and angle A = 40°; triangle 2 has e = 7, f = 9 and angle D = 40°. Angle A sits between sides b and c, and D between e and f, so the triangles are congruent by SAS. The third side is a² = 7² + 9² − 2·7·9·cos 40° ≈ 33.5, so a = 5.79 in both.

SSS, in a different order. Triangle 1 has sides 3, 4, 5. Triangle 2 has d = 5, e = 3, f = 4. Same three lengths, so SSS holds, but in a different order: a = 3 pairs with e, b = 4 with f, c = 5 with d. The pairing is A↔E, B↔F, C↔D, so the statement is △ABC ≅ △EFD, not △ABC ≅ △DEF.

ASA and AAS. Take A = 50°, B = 60° and c = 8 in both triangles. Side c joins corners A and B, so it is between the two angles: ASA. If the known side is a = 7 instead, it is opposite A and not between the angles: AAS. In both cases the third angle is 180° − 50° − 60° = 70°.

HL. Triangle 1 has hypotenuse c = 10, leg b = 8 and a right angle at C; triangle 2 has d = 10, e = 8 and a right angle at D. HL applies. The other leg is √(10² − 8²) = 6 in both.

SSA: why two sides and a non-included angle is not enough

Suppose both triangles have a = 5, b = 7 and angle A = 40°. The angle A is not between a and b; it is opposite a. The law of sines gives sin B = 7 · sin 40° ÷ 5 ≈ 0.900, and two angles have this sine: 64.15° and 115.85°. With B = 64.15° the third angle is 75.85° and c ≈ 7.54. With B = 115.85° the third angle is 24.15° and c ≈ 3.18. Two different triangles fit the same SSA data, so one triangle with these numbers need not match another.

SSA does work if the angle is opposite the longer of the two sides (only one triangle exists) or if it is a right angle (the HL case).

AAA: same shape, not necessarily the same size

Angles of 50°, 60° and 70° fix a triangle's shape but not its size: a triangle with sides of about 4.0, 4.5 and 4.9 and one with sides of about 8.0, 9.0 and 9.8 can share them, one twice the size of the other. With only angles, the calculator answers that the triangles are similar at most. Add one pair of equal sides and ASA or AAS applies; add different sides and the answer is not congruent.

How the triangle congruence solver decides

1. It checks that each triangle can exist, then fills in what follows: the third angle, angles from three sides, the third side from SAS, other sides from the law of sines.
2. It tries all six pairings of corners and drops any pairing where a known pair of parts differs.
3. Among the pairs that match it looks for SSS, SAS, ASA, AAS or HL, preferring parts you typed over computed ones. If none fits, it says whether the pattern is SSA, AAA or too little information.

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