Congruent triangles: SSS, SAS, ASA and why SSA fails

Two triangles can share a 30° angle, a side of 10 and a side of 6 and still differ: the third side is about 11.98 in one and about 5.34 in the other.

Do any three matching parts make two triangles congruent?

A triangle has six measurements, three sides and three angles. Checking all six to show two triangles are identical is tedious, so geometry looks for shortcuts: three well-chosen parts that force the other three. A common belief is that any three will do. It sounds reasonable. It is wrong twice over. Three equal angles do not fix the size, and two sides with an angle that is not between them do not always fix the shape. This unit takes both failures apart, and what remains are the conditions that work.

First, what congruent means. OpenStax says two triangles are congruent if you can pick one up and lay it on the other so that they coincide, turning or flipping it if needed. Then matching sides are equal and matching angles are equal. The matching matters. Writing △ABC ≡ △DEF promises that A goes to D, B to E and C to F, so AB = DE, BC = EF, CA = FD, and angle A = angle D and so on. Korean textbooks use the sign ≡; OpenStax and many English books write ≅ for the same idea.

Equal angles fix the shape but not the size

Take a 30°, 60°, 90° set square and a giant 30°, 60°, 90° triangle chalked on the playground. All three angles match, and nobody would call them identical. Angle-angle-angle pins down the shape only. Those two triangles are similar, the subject of the next unit, but not congruent. To fix the size, at least one length has to be part of the condition.

Two sides and an angle that is not between them

Now the subtle failure. Fix angle A at 30°, make side AB 10 units long, and ask for side BC to be 6. Swing BC around B like a compass arm. Its tip can land on the base line in two places, as the figure shows. One triangle has a third side AC of about 11.98 and angles of about 30°, 93.6° and 56.4°. The other has AC of about 5.34 and angles of about 30°, 26.4° and 123.6°. Both have a 30° angle, a side of 10 and a side of 6, and they are clearly different triangles. So side-side-angle, with the angle not between the two sides, is not a congruence condition.

Angle A is 30° and AB is 10. A circle of radius 6 around B meets the base line at C₁ and C₂, so triangle ABC₁ (orange side) and triangle ABC₂ (blue) both fit the same three facts.

There is a rescue in one case. If the side opposite the given angle is at least as long as the other given side, the swinging arm can only land once on the correct side, and the triangle is fixed. A right angle is the best-known example: when the given angle is 90°, the side across from it is the hypotenuse, the longest side, so a right triangle is fixed by its hypotenuse and one leg. Outside such cases, two sides and a non-included angle are not enough.

Three conditions that do work

Three combinations do fix a triangle completely. Three sides (SSS): if all three sides of one triangle equal the matching sides of another, the triangles are congruent. Two sides and the angle between them (SAS): the included angle locks the two sides into place, and the third side is forced. Two angles and the side between them (ASA). Two angles and some other side also work, because the angle sum already gives the third angle, and then the side sits between two known angles after all. OpenStax lists this last one separately as AAS. Notice the pattern in SAS and ASA: the angle or side that is given sits between the other two, never off to one side.

SSS	Three sides equal
SAS	Two sides and the angle between them equal
ASA	Two angles and the side between them equal
SSA	Not a condition in general: two triangles can fit
AAA	Same shape only: similar, not congruent

BackgroundIn the D. E. Joyce edition of Euclid’s Elements, the guide to Proposition 26 of Book I lists Euclid’s congruence theorems: Proposition 4 for side-angle-side, Proposition 8 for side-side-side, and Proposition 26 for a side with two angles. Joyce adds that calling them congruence theorems is a modern habit, since Euclid never used the idea of congruence by that name. He also notes that Euclid includes no side-side-angle theorem, though one special case, side-side-right angle, turns up inside the proof of Proposition 14 of Book III.

Using SAS to prove the base angles are equal

Here is the classic first proof. In isosceles triangle ABC, let AB = AC. Draw the bisector of angle A and call the point where it meets BC the point D. Compare triangles ABD and ACD. AB = AC by assumption. Angle BAD = angle CAD because AD bisects angle A. And AD is a side of both. Two sides and the included angle match, so by SAS the triangles are congruent, and the matching angles B and C are equal. Folding a paper isosceles triangle along that bisector shows the same thing; the proof explains why the fold always lines up.

Drag to turn. Scroll or pinch to zoom.

An isosceles triangle with C at the top. The angles at A and B match. Drag C sideways and they separate; drag it back above the middle of AB and they match again.

Which condition does NOT guarantee congruence in general?
- two sides and an angle not between them
- three sides
- two sides and the angle between them
- two angles and the side between them
Two triangles have angles 40°, 60°, 80° each. They are:
- similar, but not necessarily congruent
- always congruent
- never similar
- right triangles
AB = DE, BC = EF and CA = FD. Which condition shows △ABC ≡ △DEF?
- SSS
- SAS
- ASA
- none
AB = DE, angle B = angle E, BC = EF. Which condition applies?
- SAS
- SSA
- ASA
- SSS
Angle A = angle D, AB = DE, angle B = angle E. Which condition applies?
- ASA
- SAS
- SSS
- AAA
△ABC ≡ △DEF and AB = 7 cm. How long is DE?
△PQR ≡ △XYZ, angle P = 50° and angle Q = 70°. Find angle Z.
In the SSA example, A = 30°, AB = 10, BC = 6. Find the longer possible length of AC, to two decimal places.
In isosceles triangle ABC with AB = AC, the bisector of angle A meets BC at D. Which condition shows △ABD ≡ △ACD?
- SAS
- SSA
- AAA
- ASA
A right triangle has hypotenuse 13 and one leg 5. Another right triangle has the same hypotenuse and leg. Are they congruent?
- Yes
- No
- Only if the other leg is given
- Only if both are isosceles

Common mistakes

“Two sides and an angle are equal, so the triangles are congruent.” Only if the angle is between the two sides. If it is not, the compass circle can cut the base line twice and two different triangles fit, as in the 30°, 10, 6 example.
“△ABC ≡ △DEF, so AB = EF.” The order of letters is the matching. A goes with D and B with E, so AB matches DE, not EF. Write the correspondence before you read off equal parts.
“All three angles match, so they are congruent.” Equal angles give the same shape at any size. A set square and a triangle chalked on the playground can share every angle. That is similarity.

Teacher's noteIn the 2022 Korean curriculum, congruence first appears in primary grades 5 to 6 with matching points, sides and angles. The congruence conditions come in middle school, right after constructing a triangle from given parts. A later standard asks learners to justify the equal base angles of an isosceles triangle with those conditions, and that is where the word proof is introduced. The SSA case is not an achievement standard, but it is the best way to show why the angle must be included. Have learners construct the 30°, 10, 6 triangle with a compass: when two learners hold up different triangles that both follow the instructions, the need for a careful condition becomes obvious.

Key points

Congruent: one triangle fits exactly on the other; matching sides and angles are equal.
Conditions that work: SSS, SAS (included angle), ASA (included side); AAS follows from ASA.
SSA fails in general: two triangles can fit. AAA gives only similarity.
The order in △ABC ≡ △DEF tells you which parts match.

Sources for this course (7)

Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수03-08] 이등변삼각형·정삼각형, [4수03-09] 직각삼각형·예각삼각형·둔각삼각형, [4수03-25] 삼각형과 사각형의 내각의 크기의 합을 여러 가지 방법으로 추론; 초등학교 5~6학년군 [6수03-01] 합동·대응점·대응변·대응각, [6수03-14] 삼각형의 넓이(높이가 삼각형의 외부에 있는 것도 다룸); 중학교 1~3학년군 [9수03-03] 삼각형의 작도, [9수03-04] 삼각형의 합동 조건, [9수03-05] 삼각형의 내각과 외각, [9수03-09] 이등변삼각형의 성질을 합동 조건으로 정당화하고 증명이라는 용어 도입, [9수03-12]·[9수03-13] 닮음·닮음비·삼각형의 닮음 조건, [9수03-15] 피타고라스 정리를 추측하고 정당화, 역은 직관적으로 이해하고 직각삼각형 판별에 사용; 용어와 기호 대변, 대각, 내각, 외각, 증명, 닮음비, △ABC, ≡, ∽)
Textbook: OpenStax, Contemporary Mathematics, 10.3 Triangles (triangle from three noncollinear points; acute, right, obtuse; side a opposite angle A; hypotenuse and legs; angle sum 180° by a parallel line and alternate interior angles; x, x + 17, 3x − 62 gives 45°, 62°, 73°; SSS, SAS, ASA, AAS; similar triangles, scaling factor; tower shadow 180 ft with a 9 ft truck and 15 ft shadow gives 108 ft; Thales and the pyramid: openstax.org/books/contemporary-mathematics/pages/10-3-triangles)
OpenStax Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; Pythagoras around 500 BCE; 13 ft ladder 5 ft from the wall; 10 inch bracket about 7.1 inches; similar triangles: openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem)
9.4 Use Properties of Rectangles, Triangles, and Trapezoids (A = ½bh from half a rectangle; isosceles and equilateral; equilateral perimeter 93 in: .../pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids)
OpenStax Precalculus 2e, 8.2 Non-right Triangles: Law of Cosines (Heron’s formula with semi-perimeter s; sides 10, 15, 7 give s = 16 and area about 29.4; Chicago lot 62.4, 43.5, 34.1 m gives about 711.4 m²: openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines)
History for story blocks only: Euclid’s Elements, D. E. Joyce edition, Clark University (Book I Prop. 20 triangle inequality and Heron’s shortest-path problem; Prop. 26 and its guide on congruence theorems, SSA ambiguity and side-side-right angle; Prop. 32 exterior angle and angle sum, Proclus corollaries; Prop. 47 and its guide on Old Babylonians ca. 1900 to 1600 BCE, Plimpton 322, the Zhou bi hypotenuse diagram, Proclus; Prop. 48 converse; Book VI Prop. 4 equiangular triangles: mathcs.clarku.edu/~djoyce/elements/bookI/propI47.html)
MacTutor History of Mathematics, University of St Andrews, Heron of Alexandria (formula proved in Metrica Book I; eclipse of 13 March 62; side of 720 about 26 5/6: mathshistory.st-andrews.ac.uk/Biographies/Heron/) MacTutor, "Euclid of Alexandria" (born about 325 BC, died about 265 BC; dating for the Elements, about 2,300 years ago: mathshistory.st-andrews.ac.uk/Biographies/Euclid/, checked 2026-10-19).

Checked: 2026-10-19