# Geometry Circle Theorems

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## Section

Question | Answer |
---|---|

Circle | A set of all points in a plane equidistant from a given point called the center. |

Arc Addition Postulate | The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. |

If a line is tangent to a circle... | then the line is perpendicular the radius at the point of tangency. |

If a line in a plane of a circle is perpendicular to a radius at its endpoint on the circle... | then the line is tangent to a circle. |

If two tangent segments to a circle share a common endpoint outside the circle... | then the two segments are congruent. |

(central angles, arcs) In a circle or in congruent circles... (Regular) | congruent central angles have congruent arcs |

(central angles, arcs) In a circle or in congruent circles... (Converse) | congruent arcs have congruent central angles. |

(central angles, chords) Within a circle or congruent circles... (Regular) | congruent central angles have congruent chords. |

(central angles, chords) Within a circle or congruent circles... (Converse) | congruent chords have congruent central angles. |

(chords, arcs) Within a circle or in congruent circles... (Regular) | congruent chords have congruent arcs. |

(chords, arcs) Within a circle or in congruent circles... (Converse) | congruent arcs have congruent chords. |

(chords equidistant) Within a circle or in congruent circles... (Regular) | chords equidistant from the center or centers are congruent. |

(chords equidistant) Within a circle or in congruent circles... (Converse) | congruent chords are equidistant from the center (or centers). |

(parallel chords) In a circle, or in congruent circles... (Regular) | parallel chords intercept congruent arcs between them. |

(parallel chords) In a circle or congruent circles... (Converse) | congruent arcs are intercepted by parallel chords. |

In a circle, if a diameter is perpendicular to a chord... | then it bisects the chord and its arc. |

In a circle, if a diameter bisects a chord... | then it is perpendicular to the chord. |

In a circle, the perpendicular bisector of a chord contains... | the center of the circle. |

The measure of an inscribed angle is... | one half the measure of its intercepted arc. |

Two inscribed angles that intercept the same arc are... | congruent. |

An angle inscribed in a semicircle is... | a right angle. |

The opposite angles of a quadrilateral inscribed in a circle are... | supplementary. |

The measure of an angle formed by a tangent and a chord is... | half the measure of the intercepted arc. |

The measure of each angle formed by two lines that intersect inside a circle is... | one half the sum of the measures of the intercepted arcs. |

The measure of an angle formed by two lines that intersect outside a circle is... | half the difference of the measures of the intercepted arcs. |

For a given point and circle... | the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle. |

Case 1: ab = cd | The products of the chord segments are equal. |

Case 2: w(w+x) = y(y+z) | The products of the secants and their outer segments are equal. |

Case 3: t^2 = y(y+z) | The product of the secant and its outer segment equals the square of the tangent. |

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