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Math101Inscribed Angles
An inscribed angle has its vertex on a circle and sides along chords. Its measure is half the measure of its intercepted arc.
Inscribed-angle theorems infer inaccessible angles and prove cyclic relationships in geometry, surveying, and constructions.
Intuition and core definition
An inscribed angle has its vertex on a circle and sides along chords. Its measure is half the measure of its intercepted arc. Inscribed angles intercepting the same arc are congruent, and an angle subtending a diameter is a right angle (Thales’ theorem).
Notation, language, and conditions
If $\angle ACB$ intercepts arc $AB$ not containing $C$, then $m\angle ACB=\frac12m\widehat{AB}$. A central angle intercepting the same arc has twice the inscribed angle measure. Arc choice matters, especially for major and minor arcs.
Why this idea matters
An inscribed angle measures half its intercepted arc, linking a boundary vertex to a centre-based angle.
A dependable method
- Locate the angle vertex and confirm it lies on the circle.
- Identify the arc between the angle’s chord endpoints that lies inside the angle.
- Relate angle and arc by the factor $1/2$.
- Solve for the unknown and classify major/minor arc if relevant.
- Check bounds and compare with a central angle over the same arc.
