---
tag: 0GS0AW6JRY
kind: statement
role: corollary
name: Angles in the same segment are equal
msc: [51M04]
uses: [0GA1V8NN8C, 049DCJZ448, 03YN70NK0Z]
sources:
  - url: https://en.wikipedia.org/wiki/Inscribed_angle
    locator: Statement
  - url: https://en.wikipedia.org/wiki/Inscribed_angle
    locator: Applications
---

## Statement

Let $A$, $B$, $C$ and $D$ be distinct [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$. Let $\measuredangle$ be the [[oriented angle|03YN70NK0Z]], with values in $\mathbb{R} / 2\pi\mathbb{Z}$.
Then
$$2 \, \measuredangle ACB = 2 \, \measuredangle ADB.$$
For the unoriented angles: if $C$ and $D$ are on the same side of the line $AB$, then $\angle ACB = \angle ADB$. If they are on opposite sides, then $\angle ACB + \angle ADB = \pi$.

## Remarks

The statement follows from the [[inscribed angle theorem|0GA1V8NN8C]]: both sides are equal to the oriented angle $\measuredangle AOB$ at the centre $O$.
The factor $2$ is necessary. It removes the difference of $\pi$ between points on the two arcs.
The case of opposite sides says that the opposite angles of a convex cyclic quadrilateral add up to $\pi$.

## Explanations

### L1 human

A segment of a circle is the region between a chord $AB$ and one of its arcs. From every point of that arc, the chord $AB$ is seen under the same angle. From the other arc, it is seen under the supplementary angle.