Angles in the same segment are equal

Builds on Inscribed angle theorem Concyclic points Oriented angle

Statement

Let $A$ , $B$ , $C$ and $D$ be distinct concyclic points of the Euclidean plane $\mathbb{R}^2$ . Let $\measuredangle$ be the oriented angle, 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: 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 $\pi$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Angle at a point concept
Circle concept
Concyclic points concept
Oriented angle concept
Inscribed angle theorem statement
Angles in the same segment are equal statement

Tag	`0GS0AW6JRY`
URI	`m2://0GS0AW6JRY`
Kind	statement, corollary
Writing	W0 stub
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0GS0AW6JRY`.
Uses	Inscribed angle theorem, Concyclic points, Oriented angle

Angles in the same segment are equal

Statement

Remarks

Reading path

Further reading

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