How to Tell if Two Triangles Are Congruent
The short answer
Any one of those five sets of matching measurements proves two triangles are identical. Two sets that look like they should work, SSA and AAA, do not.
Congruent means identical: same shape and same size, so one could be picked up and laid exactly on the other. What makes this interesting is that you never need all six measurements. Three of the right kind are enough to pin a triangle down completely, and knowing which three is the whole subject.
List what actually matches
Write down which three measurements are equal in both, and note whether each angle sits between two known sides.
Check it against the five
SSS, SAS, ASA, AAS and RHS. If your set is one of those, the triangles are congruent.
Rule out the two impostors
An angle not between the two known sides is SSA. Three angles and no side is AAA. Neither proves congruence.
The five that work
SSS gives all three sides. Three lengths determine exactly one triangle, which is why a triangle braced on all three sides is rigid and a rectangle is not. SAS gives two sides and the angle between them, which fixes both arms and the opening, leaving the third side no choice.
ASA gives two angles and the side between them, and AAS gives two angles and a side elsewhere. Both work for the same reason: two angles fix the third, so the shape is settled, and any one side then fixes the size. RHS is a special case for right triangles, where a right angle, the hypotenuse and one other side are enough.
The two that fail
AAA fails in an obvious way once you see it. Three matching angles fix the shape but say nothing about size, so a small triangle and a large one can share all three. That is similarity, not congruence.
SSA fails in a way that is much easier to miss, and it is the reason this page exists. Two sides plus an angle that is not between them can genuinely produce two different triangles from the same three numbers.
The same three numbers, two different triangles
With a side of 6, a side of 8 and an angle of 40 degrees that does not sit between them, two genuinely valid triangles exist. Their third sides come out as 9.22 and 3.04, which are not remotely close, yet the three given measurements are identical.
a valid triangle matching all three given measurements
also valid, also matching, and a completely different shape
Why RHS escapes the problem
RHS looks like SSA at first glance, since it gives two sides and an angle that is not between them. The difference is that the angle is a right angle and the side given is the hypotenuse, the longest side of the triangle.
In the failing case, the ambiguity comes from an arc crossing the base line twice. When the angle is 90 degrees and the known side is the hypotenuse, the arc can only reach the line once, so the second solution never exists. The right angle removes the choice rather than adding information.
Why triangles are used for bracing
SSS is the reason a triangle is the shape engineers reach for. Three side lengths determine exactly one triangle, so a frame with fixed members and pinned corners cannot change shape without a member changing length.
A four sided frame has no equivalent rule. The same four lengths fold into infinitely many shapes, which is why a rectangular frame racks over unless a diagonal is added. Adding that diagonal splits it into two triangles, and the rigidity follows from SSS.
Mistakes that catch people out
- Treating SSA as a valid test. It can produce two different triangles from the same three measurements.
- Using AAA to claim congruence. It proves similarity only, and says nothing about size.
- Pairing measurements that do not correspond. The matching sides and angles have to occupy the same positions in both triangles.
- Assuming the angle in SAS can be anywhere. It has to sit between the two known sides. Move it elsewhere and you have SSA.
- Confusing RHS with SSA. RHS works only because the angle is a right angle and the side is the hypotenuse.
Practice questions
- Two triangles with sides 5, 7, 9 and 5, 7, 9. Congruent?
- Two triangles with angles 40, 60 and 80 in both. Congruent?
- Two triangles with sides 6 and 8 and an angle of 50 degrees between them. Congruent?
- Two triangles with sides 6 and 8 and an angle of 40 degrees not between them. Congruent?
- Two right triangles with hypotenuse 13 and one leg 5. Congruent?
Show the answers
- yes, by SSS, since all three sides match
- no, only similar, since AAA fixes shape but not size
- yes, by SAS, since the angle sits between the two known sides
- not necessarily, since this is SSA and two different triangles fit
- yes, by RHS, and the third side is 12 in both
Common questions
What proves two triangles are congruent?
Any one of five sets: SSS, SAS, ASA, AAS or RHS. Each fixes a triangle completely, so matching one set means the triangles are identical.
Why does SSA not work?
Because two sides and a non included angle can build two different triangles. Swinging the known side from its corner can meet the base line at two separate points.
Why does AAA not work?
Three matching angles fix the shape but not the size. That proves similarity, and a small triangle can share all three angles with a large one.
What is RHS?
A right angle, the hypotenuse and one other side. It works on right triangles because the right angle removes the ambiguity that makes SSA unreliable.
Is congruent the same as similar?
No. Congruent means identical in shape and size. Similar means the same shape at any size. Every congruent pair is similar, but not the reverse.
Do I need all six measurements to be sure?
No. Three of the right kind are enough. Checking all six is never wrong, just unnecessary.
Related lessons and calculators
- How to Tell if Two Triangles Are Similar for the looser test where only the shape has to match.
- How to Find All Three Angles When You Know All Three Sides for checking the remaining angles when only sides are given.
- How to Find the Missing Side of a Right Triangle for the RHS case worked as a calculation.
- Triangle Solver which flags the ambiguous case when the given data allows two answers.
- Area of a Triangle: Every Formula and When to Use Each for the overview of every area method in one place.
- National Council of Teachers of Mathematics for curriculum standards and how this topic is sequenced in school.
- Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.