1. What a triangle is
Three line segments connecting three points that don't all lie on the same line. The three points are the vertices; the segments between them are the sides; and at each vertex, the two sides meeting there form an interior angle.
The "don't all lie on the same line" clause — the technical word is non-collinear — is doing all the work. If your three points sit on a single line, the "segments" between them just trace and retrace that line. You'd get no enclosed region and no angles to speak of. The triangle would degenerate into a flat sliver of nothing.
A triangle with vertices $A$, $B$, $C$ is written $\triangle ABC$. It carries six pieces of data: three side lengths (call them $a$, $b$, $c$, where lowercase $a$ is the side opposite uppercase vertex $A$, and so on) and three angles ($\alpha$, $\beta$, $\gamma$, at vertices $A$, $B$, $C$).
That naming convention — side $a$ across from vertex $A$ — is universal in geometry and trigonometry. Memorize it once and never untangle "which side is which" again.
2. Classifying by sides
The first way we sort triangles is by counting how many sides have equal length.
- Equilateral — all three sides are equal. $a = b = c$.
- Isosceles — at least two sides are equal.
- Scalene — no two sides are equal. All three are different lengths.
"At least two" rather than "exactly two" matters: under most modern conventions, an equilateral triangle is a special case of an isosceles triangle — it has two equal sides, plus a bonus third. This is the same logic by which a square is a special rectangle.
A handy property: the sides and angles of a triangle pair up. Equal sides force equal opposite angles. An equilateral triangle therefore has three equal angles (each $60°$, by the angle-sum theorem in section 4). An isosceles triangle has the two angles opposite its equal sides equal — called the base angles. A scalene triangle has three different angles.
3. Classifying by angles
The second way we sort triangles is by their largest angle.
- Acute — every angle is less than $90°$.
- Right — exactly one angle is $90°$. (You can't have two: their sum would already hit $180°$, leaving nothing for the third.)
- Obtuse — one angle is greater than $90°$. (Again, only one possible.)
The two classifications — by sides, by angles — are independent. Every triangle has one label from each axis. A right isosceles triangle has a $90°$ angle and two equal legs (with angles $45°$, $45°$, $90°$). An obtuse scalene triangle has one angle past $90°$ and all three sides different. There's no equilateral right or equilateral obtuse, though — every equilateral triangle is acute, with three $60°$ angles.
4. The angle-sum theorem
One of the most-used facts in all of geometry:
$$ \alpha + \beta + \gamma = 180° $$The three interior angles of any triangle, no matter how stretched or squashed, always sum to exactly $180°$ (a straight angle). Why? The classical argument is short and beautiful — it borrows one extra line from outside the triangle and the rest falls out.
The proof in one picture
Take any triangle $ABC$. Through vertex $A$, draw a line parallel to the opposite side $BC$. Call it $\ell$.
Now look at vertex $A$. Three angles sit along the line $\ell$:
- The angle on the left is equal to $\gamma$ (the angle at $C$). Reason: $AC$ is a transversal cutting the parallel lines $\ell$ and $BC$, and these two angles are alternate interior angles.
- The angle in the middle is $\alpha$ (the original angle of the triangle at $A$).
- The angle on the right is $\beta$ (the angle at $B$) — same alternate-interior-angle argument, this time with $AB$ as transversal.
Those three angles together fill up the straight line $\ell$, so they sum to $180°$:
$$ \gamma + \alpha + \beta = 180° $$And that's exactly the claim, with the letters reshuffled. The angle-sum theorem isn't really about triangles — it's about parallel lines wearing a triangle costume.
This proof leans on the existence of a unique parallel line through $A$ — Euclid's parallel postulate. On a curved surface (the surface of a sphere, the inside of a saddle) that postulate fails and triangles don't sum to $180°$. The angles of a triangle on a globe can sum to anything from $180°$ up to $540°$. Section 6 of the pitfalls digs into this.
5. Congruence rules
Two triangles are congruent when one can be slid, rotated, or reflected onto the other so they coincide exactly. Same shape, same size, just possibly facing a different direction.
The remarkable fact is that you don't need to verify all six measurements (three sides, three angles) to confirm congruence. The right three — chosen carefully — are enough. There are four safe patterns:
| Rule | What you check | Why it works |
|---|---|---|
| SSS | All three sides equal | Three side lengths determine a triangle uniquely (up to reflection). |
| SAS | Two sides and the angle between them equal | The angle fixes how the two known sides splay apart; the third side is then forced. |
| ASA | Two angles and the side between them equal | The two angles set both directions from the known side; the other two sides meet at a forced point. |
| AAS | Two angles and a non-included side equal | Equivalent to ASA: knowing two angles tells you the third (angle-sum), so you're really back to ASA. |
Notice what's not on this list: AAA, and SSA.
AAA (all three angles equal) only proves the triangles are similar, not congruent — they have the same shape but possibly different sizes. A small equilateral and a giant equilateral both have three $60°$ angles.
SSA (two sides and a non-included angle) is the famous ambiguous case. The angle and one side determine a ray; the other side then has to reach from the end of the first side to land somewhere on that ray — and depending on its length, it might land at two different spots, one spot, or none. Without more information you can't tell which triangle is meant.
SSA is not a congruence rule. If a problem hands you two sides and an angle that isn't between them, you don't yet have enough to pin the triangle down. You need either the included angle (turning it into SAS) or another side or angle.
6. The triangle inequality
Not every triple of positive numbers makes a triangle. Try to build one with sides $1$, $2$, and $5$: lay the side of length $5$ flat, then try to attach sides of length $1$ and $2$ that meet above it. They can't — they're too short to reach each other.
The rule is:
$$ a + b > c, \qquad a + c > b, \qquad b + c > a $$For any three sides of a real triangle, each side must be strictly less than the sum of the other two. Equivalently: the longest side must be strictly less than the sum of the other two.
Why? Picture trying to walk from vertex $B$ to vertex $C$. The direct route is the side of length $a$. The detour through vertex $A$ is the sum $b + c$. The direct route is by definition the shortest path between two points, so
$$ a < b + c $$and similarly for the other two sides. The inequality is just the statement "a straight line is the shortest distance between two points" turned into algebra.
What happens at the boundary, when $a + b = c$ exactly? The two short sides line up end-to-end and lie flat along the long side. The "triangle" collapses into a single line segment — a degenerate triangle with zero area and three angles of $0°$, $0°$, $180°$. Most definitions exclude this case, which is why the inequality is strict ($>$, not $\geq$).
To check whether three lengths can form a triangle, you only need to check one inequality: the one involving the longest side. If the two shorter sides sum to more than the longest, all three inequalities are automatically satisfied.
7. Beyond the basics
A handful of further results come up constantly. None of them is hard once you have the angle-sum theorem and the congruence rules, but each one earns its own name because it gets reused so often.
The exterior angle theorem
At any vertex of a triangle, extend one side past the vertex. The angle that opens up outside the triangle, between the extension and the other side at that vertex, is called an exterior angle. The two interior angles at the other two vertices are called the remote interior angles.
The theorem is short:
An exterior angle equals the sum of the two remote interior angles.
Why? The exterior angle and its adjacent interior angle together form a straight line, so they sum to $180°$. The three interior angles also sum to $180°$. Subtract:
$$ \text{exterior} = 180° - \alpha_{\text{adjacent}} = \beta + \gamma $$That's the whole proof. As a corollary, an exterior angle is always strictly greater than either of its remote interior angles — useful for bounding arguments.
Similarity
Two triangles are similar ($\triangle ABC \sim \triangle DEF$) when they have the same shape but possibly different sizes. Equivalently: corresponding angles are equal and corresponding sides are in the same ratio:
$$ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} $$Similarity has its own short list of detection rules, parallel to the congruence rules:
- AA — two pairs of equal angles. (The third pair is then forced by the angle-sum theorem, so AAA collapses to AA.)
- SAS similarity — two pairs of sides in the same ratio with the included angle equal.
- SSS similarity — all three pairs of sides in the same ratio.
Congruence is the special case of similarity where the ratio is $1$. Every fact about congruent triangles has a similarity-scaled cousin.
The Hypotenuse-Leg rule (right triangles only)
Right triangles get one bonus congruence rule, HL (Hypotenuse-Leg): if two right triangles have equal hypotenuses and one pair of equal legs, they are congruent.
Why does this work when generic SSA doesn't? Because the right angle is fixed. The Pythagorean theorem (covered next topic) then forces the third side: if hypotenuse $c$ and leg $a$ match, the other leg $b = \sqrt{c^2 - a^2}$ is determined. SSA's ambiguity vanishes the moment the known angle is $90°$.
The two special right triangles
Two right triangles appear so often — in trigonometry, in mechanical drawings, in any problem involving regular polygons — that their side ratios are worth memorizing.
| Name | Angles | Side ratio (short : long : hypotenuse) | Comes from |
|---|---|---|---|
| $45$-$45$-$90$ | $45°$, $45°$, $90°$ | $1 : 1 : \sqrt{2}$ | Half of a square cut along its diagonal. |
| $30$-$60$-$90$ | $30°$, $60°$, $90°$ | $1 : \sqrt{3} : 2$ | Half of an equilateral triangle cut along an altitude. |
For the $30$-$60$-$90$: the side opposite the $30°$ angle is the shortest (call it $1$), the hypotenuse opposite the $90°$ angle is twice as long ($2$), and the remaining leg opposite the $60°$ angle is $\sqrt{3}$ by Pythagoras.
The four classical triangle centers
Every triangle has four canonical "center" points, each defined by three lines that meet at a single point. The concurrence isn't obvious — that all three meet is the content of each theorem.
| Center | Built from | What it is |
|---|---|---|
| Centroid | Three medians (vertex to midpoint of opposite side) | The triangle's center of mass. Divides each median in a $2{:}1$ ratio, with the longer piece on the vertex side. |
| Incenter | Three angle bisectors | Center of the largest circle that fits inside the triangle (the incircle). Equidistant from all three sides. |
| Circumcenter | Three perpendicular bisectors of the sides | Center of the unique circle through all three vertices (the circumcircle). Equidistant from all three vertices. |
| Orthocenter | Three altitudes (vertex perpendicular to opposite side) | Has no equidistance interpretation, but together with the centroid and circumcenter it lies on a single line — the Euler line. |
In an acute triangle, all four centers sit inside the triangle. In a right triangle, the circumcenter lands exactly on the midpoint of the hypotenuse and the orthocenter sits at the right-angle vertex. In an obtuse triangle, the circumcenter and orthocenter both move outside the triangle. The centroid and incenter are always interior.