Formulas for Angles in Circles Formed by Radii, Chords, Tangents, Secants
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Formulas for Working with Angles in Circles
(Intercepted arcs are arcs "cut off" or "lying between" the sides of the specified angles.)

There are basically five circle formulas that
you need to remember:                 

1.  Central Angle:  
A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.
 

Central Angle = Intercepted Arc

<AOB is a central angle.
 Its intercepted arc is the minor arc from A to B. 
m<AOB = 80º
 

Theorem involving central angles:
In a circle, or congruent circles, congruent central angles have congruent arcs.

 

2.  Inscribed Angle:
An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords.

Inscribed Angle = Intercepted Arc

<ABC is an inscribed angle.
 Its intercepted arc is the minor arc from A to C.
m<ABC = 50º

           

Special situations involving inscribed angles:

An angle inscribed in a
semi-circle is a right angle.

 

In a circle, inscribed circles that intercept the same arc are congruent.

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

The opposite angles in a cyclic quadrilateral are supplementary.
 

 

3.  Tangent Chord Angle:
 An angle formed by an intersecting tangent and chord has its vertex "on" the circle.

Tangent Chord Angle =
 Intercepted Arc

<ABC is an angle formed by a tangent and chord.
Its intercepted arc is the minor arc from A to B.
m<ABC = 60º
 

4.  Angle Formed Inside of a Circle by Two Intersecting Chords:
When two chords intersect "inside" a circle, four angles are formed.  At the point of intersection, two sets of vertical angles can be seen in the corners of the X that is formed on the picture.  Remember:  vertical angles are equal.

Angle Formed Inside by Two Chords =
 Sum of Intercepted Arcs 


Once you have found ONE of these angles, you automatically know the sizes of the other three by using your knowledge of vertical angles (being congruent) and adjacent angles forming a straight line (measures adding to 180).

 


<BED is formed by two intersecting chords.
 Its intercepted arcs are .
  [Note:  the intercepted arcs belong to the set of vertical angles.]
 
also, m<CEA = 120º (vetical angle)
m<BEC and m<DEA = 60º by straight line.

5.  Angle Formed Outside of a Circle by the Intersection of:
"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 = Difference of Intercepted Arcs 
(When subtracting, start with the larger arc.)

Two Tangents:
<ABC is formed by two tangents
intersecting outside of circle O. 
The intercepted arcs are minor arc and major arc .  These two arcs together comprise the entire circle.


Special situation for this set up:  It can be proven that <ABC and central <AOC are supplementary.  Thus the angle formed by the two tangents and its first intercepted arc also add to 180º.

Two Secants:
<ACE is formed by two secants
intersecting outside of circle O. 
The intercepted arcs are minor arcs and . 


a Tangent and a Secant:
<ABD is formed by a tangent and a secant
intersecting outside of circle O. 
The intercepted arcs are minor arcs and .