# Definition:Triangle (Geometry)/Right-Angled

## Definition

A **right-angled triangle** is a triangle in which one of the vertices is a right angle.

Note that in order to emphasise the nature of the right angle in such a triangle, a small square is usually drawn inside it.

Note the following nomenclature in the context of a **right-angled triangle**:

### Hypotenuse

In a right-angled triangle, the opposite side to the right angle is known as the **hypotenuse**.

In the above figure, the side labeled $b$ is the **hypotenuse**.

### Adjacent

In a right-angled triangle, for a given non-right angled vertex, the adjacent side which is *not* the hypotenuse is referred to as **the adjacent**.

In the above figure:

### Opposite

In a right-angled triangle, for a given non-right angled vertex, the opposite side is referred to as **the opposite**.

In the above figure:

### Legs

In a right-angled triangle, the two sides which are not the hypotenuse are referred to as its **legs**.

## Euclid's Definition

In the words of Euclid:

*Further, of trilateral figures, a***right-angled triangle**is that which has a right angle, an**obtuse-angled triangle**that which has an obtuse angle, and an**acute-angled triangle**that which has its three angles acute.

(*The Elements*: Book $\text{I}$: Definition $21$)

## Also known as

A **right-angled triangle** is also often referred to as a **right triangle**.

## Also see

- Results about
**right triangles**can be found**here**.

## Sources

- 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**triangle**(Euclidean geometry) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**triangle** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**triangle** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**right-angled triangle**