# Angle

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

In elementary Euclidean geometry, an angle is formed by two rays with a common endpoint, and its opening can be described in degrees.

mko-angle · concept · v0.10.0



## Mathematical statement

Two rays with a common endpoint form an angle. The common endpoint is the vertex and the rays are the sides; this object treats unoriented angles from 0° through 180° measured in degrees.





## Representative formula (exact TeX)

\theta





## Assumptions


- The setting is elementary Euclidean geometry with degree measure, restricted to unoriented angles from 0° through 180°.





## Symbol scope


- θ — angle or its explicitly stated measure





## Intuition, definition, and misconceptions

The size of an angle is determined by the opening between its sides, not by how long or thick the rays are drawn. Angles with 0°<θ<90°, θ=90°, 90°<θ<180°, and θ=180° are called acute, right, obtuse, and straight angles respectively.

Two angles are complementary when their measures sum to 90° and supplementary when they sum to 180°. These relations depend on addition of measures and do not require the angles to be adjacent in a diagram.

This object fixes elementary unoriented angles. Directed rotation, coterminal angles, radians, and angles in inner-product spaces require additional directional, periodic, or algebraic structure.

Common mistakes include guessing a measure from a sketch, treating side length as angle size, or assuming complementary and supplementary angles must be adjacent.





## Adopted derivation guide

[Angle: a step-by-step proof that vertical angles are equal](https://ocme.evemisslab.com/en/derivations/derivation-mko-angle-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Points, lines, and planes](https://ocme.evemisslab.com/en/mko/mko-point-line-plane/) — The vertex and two rays of an angle use the language of foundational geometric objects.





## Verification boundary

Explicit degree values support finite arithmetic recomputation, but they do not formally prove properties of angles, rays, or angle measure.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Choose a EuclideanGeometry or inner-product-space model and bind rays, vertex, angle measure, and degree conversion precisely.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/angle.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-angle/index.md)