1. What an angle is
The figure formed by two rays — called the sides of the angle — that share a common endpoint, called the vertex. The angle itself is the rotation needed to sweep one ray onto the other.
Three ingredients, no more: a vertex, and two rays leaving it. The picture below is the entire idea. Everything else on this page — units, types, pairs, parallel-line theorems — is built on top of those three things.
Notation
Three common ways to name the same angle above:
- $\angle AVB$ or $\angle BVA$ — three letters, with the vertex letter in the middle. This is unambiguous and is the form to use when more than one angle shares a vertex.
- $\angle V$ — one letter, when only one angle lives at that vertex.
- $\theta$, $\alpha$, $\beta$ — a single Greek letter, labelled directly on the figure.
In three-letter angle notation, the middle letter is always the vertex. $\angle AVB$ and $\angle BVA$ both mean the angle at $V$. $\angle VAB$ is a different angle — its vertex is $A$.
2. Measuring angles in degrees
An angle is a measurement of rotation. Pick a starting ray, rotate it until it lands on the second ray, and however much you rotated — that's the angle. To compare angles you need a unit. The classical one is the degree.
A full revolution — rotating all the way back to where you started — is divided into $360$ equal slices. One slice is one degree.
$$ 1 \text{ full turn} = 360^\circ $$The number 360 is historical rather than mathematical. It survives because it has an astonishing number of divisors — $1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360$ — so most of the angles you actually want come out to whole numbers.
Reference angles to memorize
| Rotation | Degrees | Looks like |
|---|---|---|
| No rotation | $0^\circ$ | The two rays coincide. |
| Quarter turn | $90^\circ$ | A right angle — the rays are perpendicular. |
| Half turn | $180^\circ$ | A straight line — the rays point in opposite directions. |
| Three-quarter turn | $270^\circ$ | Three right angles stacked. |
| Full turn | $360^\circ$ | Back to no rotation. |
Degrees are convenient for humans, but most of mathematics (and all of calculus) prefers radians, defined so that a full turn is $2\pi$. The conversion is $180^\circ = \pi \text{ rad}$. We use degrees on this page; radians get their own topic.
3. Types of angles
Geometry gives every range of degrees a name. The names come up constantly — being fluent in them is half the vocabulary of the subject.
| Name | Range | Mnemonic |
|---|---|---|
| Zero | $=0^\circ$ | The two rays coincide. |
| Acute | $0^\circ < \theta < 90^\circ$ | "A-cute little angle." |
| Right | $=90^\circ$ | The corner of a square. Marked with a small square, not an arc. |
| Obtuse | $90^\circ < \theta < 180^\circ$ | Wider than a right angle but not yet a straight line. |
| Straight | $=180^\circ$ | Sides form a straight line through the vertex. |
| Reflex | $180^\circ < \theta < 360^\circ$ | The "outside" angle — the long way around. |
| Full | $=360^\circ$ | A complete revolution. |
A right angle is conventionally marked with a small square at the vertex rather than an arc. When you see that square, you can use $90^\circ$ in your calculation without re-checking.
4. Angle pairs
Angles rarely appear in isolation. The geometry of figures — triangles, parallelograms, polygon nets — gets most of its mileage from relationships between two angles. Four such relationships come up so often they have names.
Complementary angles
Two angles whose measures add to $90^\circ$:
$$ \alpha + \beta = 90^\circ $$Either angle is called the complement of the other. They needn't be adjacent — "complementary" is a statement about the sum, not the picture. If $\alpha = 35^\circ$, its complement is $55^\circ$ whether the two angles touch or live on opposite ends of the page.
Supplementary angles
Two angles whose measures add to $180^\circ$:
$$ \alpha + \beta = 180^\circ $$Either angle is the supplement of the other. The supplement of $35^\circ$ is $145^\circ$.
Linear pair
A pair of adjacent angles that together form a straight line — meaning they share a vertex, share a ray, and their other rays point in exactly opposite directions. Every linear pair is supplementary; the picture below shows why.
Vertical angles
When two lines cross, they create four angles in an X pattern. The angles directly across the vertex from each other are vertical angles, and the key fact about them is:
Vertical angles are equal.
The proof is one line of algebra. Call the four angles $\alpha, \beta, \alpha', \beta'$ going around. The two adjacent pairs $(\alpha, \beta)$ and $(\beta, \alpha')$ are each linear pairs, so:
$$ \alpha + \beta = 180^\circ \quad \text{and} \quad \beta + \alpha' = 180^\circ $$Subtracting gives $\alpha = \alpha'$. The same argument with the other pair shows $\beta = \beta'$.
A common pre-test mix-up. Complementary sums to a corner ($90^\circ$). Supplementary sums to a straight line ($180^\circ$). The letter c in "complementary" matches the c in "corner," and the s in "supplementary" matches the s in "straight." Pick a mnemonic and stick to it.
Adjacent angles
Two angles are adjacent when they share a vertex and a side but their interiors do not overlap. A linear pair is the special case of adjacent angles whose non-shared sides form a straight line. Vertical angles, by contrast, are not adjacent — they share only the vertex.
Angle addition postulate
If point $C$ lies in the interior of $\angle AVB$, then the two adjacent angles $\angle AVC$ and $\angle CVB$ add to the whole:
$$ m\angle AVC + m\angle CVB = m\angle AVB $$This is the rule that lets you split a big angle into pieces, or combine pieces into a bigger angle. It's the angle analogue of "the whole equals the sum of its parts" and underpins most multi-step angle calculations.
Angle bisector
A ray that splits an angle into two equal halves is called the angle bisector. If ray $VC$ bisects $\angle AVB$, then by combining the bisector definition with the angle addition postulate:
$$ m\angle AVC = m\angle CVB = \tfrac{1}{2}\, m\angle AVB $$So the bisector of a $70^\circ$ angle produces two $35^\circ$ angles. Every angle has exactly one bisector.
5. Parallel lines and a transversal
Draw two lines, then draw a third line that crosses both of them. The third line is called a transversal. At each crossing it creates four angles, so there are eight angles in the figure — and a tidy vocabulary for how those eight are related.
When the original two lines are parallel, the eight angles collapse into just two distinct values: every angle is either equal to a chosen reference angle, or supplementary to it. That collapse is the most useful single fact in elementary geometry.
The four named pairings
Using the numbering in the diagram:
| Pair type | Examples | When the lines are parallel |
|---|---|---|
| Corresponding | (1, 5), (2, 6), (3, 7), (4, 8) | Equal: $\angle 1 = \angle 5$, etc. |
| Alternate interior | (3, 5), (4, 6) | Equal: $\angle 3 = \angle 5$. |
| Alternate exterior | (1, 7), (2, 8) | Equal: $\angle 1 = \angle 7$. |
| Co-interior (same-side interior) | (3, 6), (4, 5) | Supplementary: $\angle 3 + \angle 6 = 180^\circ$. |
- Corresponding means "same corner at each intersection" — both upper-right, or both lower-left, etc.
- Interior angles live between the two parallel lines; exterior angles live outside them.
- Alternate means on opposite sides of the transversal; co- (or same-side) means on the same side.
The "if and only if" that powers everything
The headline theorem — the one you'll actually use — is a biconditional:
Two lines cut by a transversal are parallel if and only if corresponding angles are equal.
Both directions are useful. The "$\Rightarrow$" direction says: if you know the lines are parallel, you may conclude angles are equal — that's how you compute unknown angles in figures. The "$\Leftarrow$" direction says: if you can show one pair of corresponding angles is equal, you may conclude the lines are parallel — that's how you prove parallelism. Each of the other three pair-types (alternate interior, alternate exterior, co-interior) gives an equivalent biconditional. Pick whichever one is convenient.
At each intersection, vertical angles are equal and linear pairs are supplementary — so only two distinct values appear at each crossing. The parallel-lines theorem then says the values match across the two intersections. Two distinct values, total. If you've solved for one, you've solved for all eight.