Circle Angles: Central, Inscribed and Tangent Angles UPCAT Reviewer
Circle Angles
Recognize which arc an angle intercepts and choose the correct circle relationship before calculating.
Circle Geometry - Angles
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Circle angles are controlled by intercepted arcs
A central angle has its vertex at the center of the circle. An inscribed angle has its vertex on the circle. Both angles below intercept the same highlighted arc AB.
Central and inscribed
For the same arc, the central angle is twice the inscribed angle.
Radius and tangent
A radius to a point of tangency is perpendicular to the tangent.
Find the vertex, then find the arc
Why it works
The vertex location determines how the angle is built from radii or chords. The intercepted arc identifies the exact portion of the circle controlling its measure.
Five forms you should recognize
A 100° arc gives a 100° central angle.
A 100° arc gives a 50° inscribed angle.
A 32° inscribed angle and its central angle correspond as 32° and 64°.
The angle between a tangent and radius at contact is 90°.
If one interior angle is 112°, the opposite angle is 68°.
Check before you commit
- Doubling the wrong way: inscribed angle is half its arc.
- Wrong intercepted arc: use the arc between the angle’s chord endpoints.
- Assuming every chord is a diameter: look for the center or diameter mark.
- Missing the right angle: radius and tangent are perpendicular.
- Adjacent vs opposite: only opposite cyclic-quadrilateral angles are supplementary.
- Diagram dependence: do not trust visual scale; use stated relationships.
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Circle Angles FAQ
How do I identify an inscribed angle?
Its vertex lies on the circle and its sides are chords.
What angle subtends a diameter?
An inscribed angle intercepting a semicircle is 90°.
Are all central angles twice all inscribed angles?
Only when they intercept the same arc.
Why are opposite cyclic angles supplementary?
Their intercepted arcs together make the full 360° circle, and each angle is half its arc.
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