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Math101learn.math101.caInscribed 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.
Worked example
Representations and interpretation
Moving an inscribed-angle vertex along the same arc leaves the intercepted opposite arc unchanged, so the angle measure stays constant. A central angle at the centre opens across the same endpoints with twice the rotation.
Reasoning about variations
The half-arc rule does not apply unchanged to angles with vertex inside or outside the circle; those use sums or differences of arcs. First classifying the vertex prevents theorem confusion.
Common mistakes
How to check your work
- Double the angle and compare with the intercepted arc measure.
- For a diameter arc of $180^\circ$, verify the angle is $90^\circ$.
- Use supplementary opposite angles in a cyclic quadrilateral as a cross-check.
Practice
- An inscribed angle intercepts an $84^\circ$ arc. Find the angle.
- What angle is subtended by a diameter at the circle?
- Two inscribed angles intercept the same arc. How are they related?
Answers and brief solutions
Show answers
- $42^\circ$ An inscribed angle is half its intercepted arc.
- $90^\circ$ It intercepts a semicircle of $180^\circ$.
- They are congruent Both equal half the same arc measure.
Synthesis and transfer
Camera viewpoints placed along the same arc see a fixed chord under equal inscribed angles, while moving to the opposite arc changes which arc is intercepted.
If chord $AB$ subtends an $80^\circ$ minor arc, every inscribed angle with vertex on the opposite major arc intercepting that same minor arc measures $40^\circ$. A vertex placed on the minor arc intercepts the major arc instead and produces a different obtuse angle, so the intercepted arc must be named rather than guessed. When $AB$ is a diameter, the intercepted semicircle is $180^\circ$, giving a right angle; this is Thales' theorem. The centre angle over a fixed arc is twice the corresponding inscribed angle, offering a construction-based verification independent of visual size.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
An inscribed angle intercepts an $84^\circ$ arc. Find the angle.
- An inscribed angle is half its intercepted arc.
End of lesson
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