Formulas for Angles in Circles MathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use   Contact Person: Donna Roberts

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

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 In the diagram at the right, ∠AOB is a central angle with an intercepted minor arc from A to B. m∠AOB = 82º In a circle, or congruent circles, congruent central angles have congruent arcs. (the converse is also true) In a circle, or congruent circles, congruent central angles have congruent chords. (the converse is also true)

 

2. Inscribed Angle An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords. Inscribed Angle = Intercepted Arc In the diagram at the right, ∠ABC is an inscribed angle with an intercepted minor arc from A to C. m∠ABC = 41º An angle inscribed in a semicircle is a right angle. (Called Thales Theorem.) The opposite angles in a cyclic quadrilateral are supplementary. A quadrilateral inscribed in a circle is called a cyclic quadrilateral. ∠x and ∠y are supplementary In a circle, inscribed angles that intercept the same arc are congruent.

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

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 = (SUM of Intercepted Arcs) 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. also, m∠BEC = 43º (vertical angle) m∠CEA and m∠BED = 137º by straight angle formed. 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 = (DIFFERENCE of Intercepted Arcs) Two Tangents: ∠ABC is formed by two tangents intersecting outside of circle O. The intercepted arcs are major arc and minor arc . These two arcs together comprise the entire circle. Angle Formed by Two Tangents = (DIFFERENCE of Intercepted Arcs) (When subtracting, start with the larger arc.) 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 Two Secants: ∠CAE is formed by two secants intersecting outside of circle O. The intercepted arcs are major arc and minor arc . Angle Formed by Two Secants = (DIFFERENCE of Intercepted Arcs) (When subtracting, start with the larger arc.) a Tangent and a Secant: ∠BAD is formed by a tangent and a secant intersecting outside of circle O. The intercepted arcs are arc and arc . Angle Formed by Tangent and Secant = (DIFFERENCE of Intercepted Arcs) (When subtracting, start with the larger arc.)

 

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