Formulas For Angles In Circles - MathBitsNotebook(Geo - CCSS Math)

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GeometrySmallLogo Condensed List of Circle Angle FormulasMathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts divider
For more in-depth information about each of these angles see Circles. circleformulasbanner
1. Central Angle A central angle is an angle formed by two radii with the vertex at the center of the circle.
Central Angle = Intercepted Arc rules1m
In the diagram at the right, ∠AOB is a central angle with an intercepted minor arc from A to B. 		angles1 m∠AOB = 82º
2. Inscribed AngleAn inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords.
Inscribed Angle = rule2mIntercepted Arc angle2m
In the diagram at the right, ∠ABC is an inscribed angle with an intercepted minor arc from A to C. 		angles1 m∠ABC = 41º
3. Tangent Chord AngleAn angle formed by an intersecting tangent and chord has its vertex "on" the circle.
Tangent Chord Angle = rule2mIntercepted Arc angletancm
In the diagram at the right, ∠ABC is an angle formed by a tangent and chord with an intercepted minor arc from A to B. 		angletancm∠ABC = 74º
4. Angle Formed by Two Intersecting Chords
When two chords intersect inside a circle, four angles are formed. At the point of intersection, two sets of congruent vertical angles are formed in the corners of the X that appears.
Angle Formed by Two Chords = rule2m(SUM of Intercepted Arcs) anglechordm
angleChord m∠AED = 43º
Also, m∠BEC = 43º (vertical angle) m∠CEA and m∠BED = 137º by straight angle formed.
5. Angle Formed Outside of Circle by Intersection: "Two Tangents" or "Two Secants" or a "Tangent and a Secant".
The formulas for all THREE of these situations are the same: Angle Formed Outside = rule2m(DIFFERENCE of Intercepted Arcs)
Two Tangents:
∠ABC is formed by two tangents intersecting outside of circle O. The intercepted arcs are major arc ac and minor arc ac. These two arcs together comprise the entire circle.
Angle Formed by Two Tangents = rule2m(DIFFERENCE of Intercepted Arcs) angles2tansMB (When subtracting, start with the larger arc.)
angle2tans NewAngle m∠ABC = 60º
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Two Secants:
∠CAE is formed by two secants intersecting outside of circle O. The intercepted arcs are major arc ceand minor arc bd.
Angle Formed by Two Secants = rule2m(DIFFERENCE of Intercepted Arcs) angleSecM (When subtracting, start with the larger arc.)

ang2sec angleSecm2 m∠CAE = 35º

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a Tangent and a Secant:
∠BAD is formed by a tangent and a secant intersecting outside of circle O. The intercepted arcs are arc bcand arc bd.
Angle Formed by Tangent and Secant = rule2m(DIFFERENCE of Intercepted Arcs) angSTm (When subtracting, start with the larger arc.)

ang2sec angSTm2 m∠BAD = 47º

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