Triangle congruence criteria: how to write the proofs

It is the end of September 2026: at the liceo classico (the classical high school), with 3 hours of maths a week, the chapter on triangles and their congruence criteria arrives between November and December, right after the primitive entities (points, lines, planes), segments and angles. The first test that contains a real proof — not just calculation — therefore falls between December and January: from here you have between nine and twelve weeks, the right time to get there without the “I don’t know where to start” anxiety that I hear repeated every year from the same students. At the science liceo, the language liceo and the human sciences course the programme is identical, only the number of weekly hours changes and so the pace at which you get there.

The problem, almost always, is not geometry itself: it is that a proof asks for a kind of reasoning that was never done in full in lower secondary school. Here you will find the three criteria with their precise statement, the scheme for setting up any proof, the mistakes that cost marks and three proofs worked step by step.

What it means for two triangles to be congruent

Two figures are congruent when they overlap perfectly, point by point: same shape, same size. For two triangles this happens when they have, in order, congruent three sides and three angles — six conditions. The discovery that makes Euclidean geometry manageable is that you don’t need to check all six: three well-chosen conditions are enough, and the other three follow. These are the three criteria.

First criterion (SAS). Two triangles are congruent if they have two sides and the angle between them congruent.

Second criterion (ASA). Two triangles are congruent if they have one side and the two angles adjacent to it congruent.

Third criterion (SSS). Two triangles are congruent if they have all three sides congruent.

The first and second criteria are taken as postulates in the Euclidean geometry studied at high school: they are not proved, they are used. The third is proved from the first two, but in the practice of a test you treat it like the other two: you check the conditions and apply it. (The Italian textbooks name them by lati and angoli, LAL, ALA, LLL; here I use the usual English abbreviations.)

A point that causes confusion every year: there is no “SSA criterion”. Two sides and an angle that is not between them are not enough to guarantee congruence — there are pairs of triangles with that condition that are not congruent. If in an exercise you have two congruent sides and a congruent angle but the angle is not the one between the two sides, that is not a valid application of any criterion: you have to find another way, usually through one more angle or side that can be derived from the data.

The scheme for setting up a proof

A proof is frightening because it seems like free text, when it is exactly the opposite: it is the most rigid form of all first-two-years maths. It always has three parts, always in the same order.

Hypothesis (given). What the text gives you as true, translated into symbols. If the text says ”$M$ is the midpoint of $AB$”, the hypothesis is written $AM\cong MB$. If it says “the bisector of angle $A$”, the hypothesis is that that angle is divided into two congruent parts. Translating the hypothesis well is half the work: a badly translated hypothesis makes the proof impossible, however good you are at the later steps.

Thesis (to prove). What you have to prove, also translated into symbols. If the text asks “prove that the two triangles are congruent”, the thesis is $△ABC\cong △A^{\prime }{B}^{\prime }{C}^{\prime }$; if it asks “prove that the two angles are equal”, the thesis is that specific congruence, not the whole triangle.

Two-column proof. Each line has a statement on the left and its justification on the right — a datum of the hypothesis, a definition, an already-proved theorem, or a congruence criterion. No line may appear without a stated justification: it is the rule that distinguishes a proof from a commented drawing.

Everything else follows from this table, for a reason worth understanding once and for all: if two triangles are congruent, every pair of corresponding elements is congruent — not only the three sides and three angles used to prove it, but also corresponding medians, bisectors, altitudes. It is the move that unlocks most exercises: you prove the triangles congruent, and the real thesis (often about an angle or a segment that seems unrelated) follows automatically.

The mistakes that cost the most marks

1. Using the thesis as if it were a hypothesis. It is the gravest and most frequent mistake: writing among the statements exactly what you should be proving, taking it for granted. If you notice you wrote the thesis halfway through a proof, go back: the line doesn’t hold.

2. “You can see it from the drawing”. No statement can rest on the drawing. The drawing helps you understand, it is not proof: two segments that look equal in the drawing may not be equal in the hypothesis, and a marker puts red on every line that doesn’t have an explicit justification.

3. The ASA criterion with the wrong side. ASA requires the congruent side to be between the two congruent angles, that is, adjacent to both. If you have two congruent angles and a congruent side but that side is not between the two angles, it is not a valid application — even if by eye it “seems” the same case.

4. Mixing up the order of the vertices in the congruence. Writing $△ABC\cong △DEF$ is not a stylistic choice: it declares that $A$ corresponds to $D$, $B$ to $E$, $C$ to $F$. If in the proof the sides and angles you have checked correspond in a different order, the writing of the congruence has to be corrected accordingly — otherwise the conclusions you draw from it (which side corresponds to which) are wrong even if the congruence itself is true.

5. The SSA criterion that doesn’t exist. It is worth repeating: two sides and a non-included angle are not enough. If you are missing exactly that condition, look in the hypothesis (or in a theorem already seen, such as the base angles of an isosceles triangle) for the extra element you need to use SAS, ASA or SSS in the correct form.

Three worked proofs

1. The isosceles triangle: the bisector is also a median and an altitude

Hypothesis: triangle $ABC$ with $AB\cong AC$ (isosceles with apex $A$); $AD$ the bisector of angle $\hat{A}$, with $D$ on $BC$.

Thesis: $\hat{B}\cong \hat{C}$ (base angles congruent); $BD\cong DC$ ($AD$ is also a median); $AD\perp BC$ ($AD$ is also an altitude).

The central point of this proof, the one that makes it a case to remember by heart: the common side $AD$ is not a datum of the hypothesis, and yet it is a perfectly valid justification — a segment is always congruent to itself. It is the first time a student meets this move, and from here on it comes back in almost every proof with a symmetric figure.

2. Midpoint and angles

Hypothesis: $M$ is the midpoint of segment $AB$; from $M$ start segments $MC$ and $MD$, on opposite sides of $AB$, with $MC\cong MD$ and $\angle AMC\cong \angle BMD$.

Thesis: $AC\cong BD$.

Here the two triangles don’t share a side: they share a vertex, $M$, and the congruence of the angles has to be looked for in the hypothesis, not deduced from an automatic geometric property (they are not vertically opposite angles in the strict sense, because $C$ and $D$ are not necessarily on the extension of each other: that is why the hypothesis gives it to you explicitly). It is the right exercise for training to distinguish what is given from what would have to be proved separately.

3. Isosceles trapezium: the diagonals are congruent

Hypothesis: trapezium $ABCD$ with $AB\parallel CD$, isosceles, that is, with congruent legs $AD\cong BC$ and congruent base angles $\angle DAB\cong \angle CBA$.

Thesis: the diagonals are congruent, $AC\cong BD$.

This is the proof “with overlapping sides”: the two triangles $DAB$ and $CBA$ are not separate in the drawing, they overlap on the same segment $AB$, which serves as a side for both but with the vertices swapped ($A$ and $B$ play opposite roles in the two triangles). It is the point where most students get lost, because the drawing shows only one trapezium and you need to imagine the two triangles “inside” the same figure instead of two separate triangles. Rewriting the congruence respecting the order of the vertices — $△DAB\cong △CBA$, not $△ABD\cong △ABC$ — is what makes the right thesis come out: $BD$ corresponds to $AC$ only if the vertices are written in that order.

How to train over the coming weeks

There is no shortcut other than doing plenty, but with a precise method: for every new exercise, before looking at the solution, write the hypothesis and thesis in symbols yourself. It is the step that, done well, solves half the exercises on its own — because once the hypothesis and thesis are written clearly, the criterion to use often jumps out at you.

A mistake to avoid: don’t learn the textbook proofs by heart word for word. If even just the names of the vertices or the orientation of the figure change, a proof learned by heart and not understood stalls at the first different step. Study the scheme — hypothesis, thesis, criterion — not the text.

To train with closed-answer questions on the whole block (primitive entities, angles, triangles and criteria) there is the Year 1 Euclidean geometry practice test: 15 questions in 50 minutes, with a breakdown of weak points by topic. The correct answer to each question is free and needs no registration; the guided step-by-step solution unlocks with your first lesson. If your test also includes the more algebraic part of the October programme, you will find it in the practice test on numbers, fractions and powers.

If you came here because the first test with calculations — not yet proofs — has already put you in difficulty, I wrote a dedicated guide in the first maths test in Year 1; and if the chapter on sets and logic seems disconnected from everything else, it isn’t: the implication “if… then…” that you study there is exactly the hypothesis-thesis structure of every proof, and it is explained in full in sets and logic in Year 1. For a revision of the whole year in a single document there is the Year 1 maths formula sheet.

If proofs remain the weak point even after understanding the scheme, the free introductory call is for exactly this: you send me two or three proofs from your notebook and I tell you honestly where the thread gets lost. Maths lessons are online and one to one: you start with a free 30-minute introductory call.