Note: The term "intercepted arc" refers to an arc "cut off" or "lying between" the sides of the specified angle.

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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.
m
AOB = 82º

angles1
theorem In a circle, or congruent circles, congruent central angles have congruent arcs.
(the converse is also true)
arcpic1
arcm1
theorem In a circle, or congruent circles, congruent central angles have congruent chords. (the converse is also true)
arcpic2
arc2m

 

2. Inscribed Angle
An 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.
m
ABC = 41º

angles1
theorem An angle inscribed in a semicircle is a right angle. (Called Thales Theorem.)
anglesemi
anglesimim
theorem The opposite angles in a cyclic quadrilateral are supplementary.
anglesemi

A quadrilateral inscribed in a circle is called a cyclic quadrilateral.

angcycm
x and ∠y are supplementary
theorem In a circle, inscribed angles that intercept the same arc are congruent.
anglesame anglesamem



3. Tangent Chord Angle
An 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.
m
ABC = 74º

angletanc



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
In the diagram at the right, ∠AED is an angle formed by two intersecting chords in the circle. Notice that the intercepted arcs belong to the set of vertical angles.angleChordm2
also, mBEC = 43º (vertical angle)
mCEA and mBED = 137º by straight angle formed.


angleChord


Once you have found ONE of these angles, you automatically know the sizes of the other three by using vertical angles (which are congruent) and adjacent angles forming a straight line (whose measures add to 180º).

 

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
Note: It can be proven that ∠ABC and central angle ∠AOC are supplementary.
Thus the angle formed by the two tangents and the degree measure of the first minor intercepted arc also add to 180º

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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


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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

 


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