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# Theorems/Postulates

### Geometry Theorems and Postulates

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Addition Property of Equality | For real numbers a,b, and c, if a=b, then a+c=b+c |

Additive Indenity | The sum of any real number and zero is that same real number. In other words, for any real number a, a+0=a |

Alternate Interior Angles Theorem | If a transversal intersects two parallel lines, then alternate interior angles are congruent |

Alternate Exterior Angles Theorem | If a transversal intersect two parallel lines, then alternate interior angles are congruent. |

Angle Addition Postulate | Thye measure of an angle created by two adjacent angles can be found by adding the measures of the two adjacent angles. |

Angle-Angle(AA) Similarity Postulate | If two corresponding angles of two or more triangles are congruent, the triangles are congruent |

Angle-Angle-Side (AAS) Postulate | If two angles and a non-included side are congruent to the corresponding two angles and a side of a second triangle, then the two triangles are congruent. |

Angle-Angle Similarity Postulate | Two corresponding angles of two or more triangles are congruent, the triangles are similar. |

Angle-Angle-Side (AAS) Postulate | If two angles and a non-included side are congruent to the corresponding two angles and a side of another triangle, the two triangles are congruent. |

Angle-Side-Angle (AAS) Postulate | If two angles and the included side of one triangle are congruent to two angles and an included side of another triangle, the triangles are congruent. |

Arc Addition Postulate | The measure of an arc created by two adjacent arcs can be found by adding the measures of two adjacent arcs |

Area of a Square | The area of the square is a measurement representing the spacer within the interior of a square. It is found by the formula A=Ssquared where a is the area and s is the length of aa side. |

Area of a Triangle | The area of a triangle is a measurement representing the space within the interior triangle. It is found by the formula A=1/2bh where a is the area, b is the base, and h is the length of the height. |

Assoiative property of Addition | For all real numbers, a,b, and c (a+b)+c equals (a+b)+C. |

Assoiative Property of Multiplication | For all real numbers, a, b, and c, a+(b+c)=(a+b)+c. |

Associative Property of Multiplication | For all real number a, b, and c, a times (b times c). |

Commutative property of Addition | For all real numbers a, b, and c, A+(b + c) equals (a + b) + c. |

Associative Property of Multiplication | For all real numbers a, b, and c, a time(b times c) equals (a time b) times c. |

Communiative Property of Multiplication | For all real numbers a and b, a times b equals b times a |

Congruent Arcs and Chords Theorem | Two minor arcs within a the same circle or between congruent circles are congruent if snd only if their corresponding chords are congruent. |

Congruent Inscribed Angles Theorem | Two or more inscribed angles that intercept the same arc, or congruent arcs, are congruent. |

Corresponding Angles Postulate | If a transversal intersects two parallel lines then the corresponding parts are congruent. |

Cross Product Property | For real numbers a, b, c and d, a divided by b equals c divided by d is equivalent to a tomes d equals b times c or ad equals bc. |

Distance Between Two Points Postulate | The distance between two points can be found by taking the absolute value of the difference between the coordinates of the two pints. |

Distributive Property | For all real numbers a, b, and c, a(b+c)= ab + ac. |

Division Property of Equality | For real numbers a, b, and c, if a=b and c is not equal to 0, then a divided by d equals b divided by c. |

Exterior Angle to a Circle Theorem | If two secants, two tangents, or a secant or a tangent intersect outside of a circle, the angle created between them is one half the absolute value of the difference of the measures of their intercepted arcs. |

Hypotenuse-Leg (HL) Theorem | If two right triangles have congruent hypotenuses and corresponding, congruent legs, the two right triangles are congruent. |

Identity Property of Division | Any number divided by one results in the same number, for example c divided by one equals c. |

Inscribed Angle to a Semicircle Theorem | An inscribed angle that intersects a semicircle is a right angle. |

Inscribed Angle Theorem | The measure of an inscribed angle is equal to one half the measure of its intercepted arc |

Inscribed Quadrilateral Theorem | The opposite angles of an inscribed quadrilateral to a circle are supplementary. |

Intersecting Chords Theorem | The point of intersection between two chords in a circle creates two pairs of segments whose products are equal. |

Intersecting Lines Postulate | If two lines intersect, then they intersect at exactly one point. |

Isosceles Triangle Theorem | If two sides are congruent, then the angles opposite those sides are congruent. |

The Converse of the Isosceles Triangle Theorem | If two angles of a triangle are congruent, then the sides opposite those angles are congruent making the triangle an isosceles triangle. |

Midsegmeent of a Triangle Theorem | A segment connecting the two sides of a triangle is parallel to the third side and half its length. |

Multiplication Property of Equality | for real numbers a, b, and c, if a=b then ac = bc. |

Multiplicative Identity | Multiplying any number by 1 produces that number, for example b times 1 equals b |

Opposite Angle Theorem | If two angles of a triangle are not congruent, then the larger side lies opposite the larger angle. |

Parallel Postulate | Given a line and a point not on that line, there exists only one line through the given point parallel to the given line. |

Perpendicular Diameter and Chords Theorem | If a diameter is perpendicular to a chord, then the diameter bisects the chord and thee minor arc between the endpoints of the chord |

Pieces of Right Triangles and Similarity Theorem | If a altitude is drawn from the right angle of a right triangle is the geometric mean between the segments of the hypotenuse created by the intersection of the altitude and the hypotenuse. |

Second Corollary to the Pieces of Right Triangles Similarity Theorem | Each leg of a right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse created by the altitude adjacent to the given leg. |

Pythagorean Theorem | If a right triangle has sides a and b and hypotenuse c then a squared plus b squared equals c squared. |

Converse of the Pythagorean Theorem | In a triangle with sides a, b, and c if a squared plus b squared equals c squared, the triangles is a right triangle. |

Reflexive Property of Equality | for a real number a, a equals a |

Same-side interior angles | If a transversal intersects two parallel lines then the same-side interior angles are supplementary. |

Converse of the same-side interior angles theorem | If the same-side interior angles formed by two lines and a transversal are supplementary, then the two lines are parallel. |

Secant-Tangent Intersection Theorem | When a secant and a tangent intersects at the point of tangency, the angles created at the point of intersection are half the measurement of the arcs they intersect. |

Segment Addition Postulate | If point C is between points A and B, then AC plus CB equals AB. |

Side-Angle-Side (SAS) Postulate | If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent. |

Side-Angle-Side Similarity Postulate | If two or more triangles have corresponding, congruent angles and the sides that make up these angles are proportional, then the triangles are similar. |

Side-Side-Side (SSS) Postulate | If the sides are congruent to the sides of a second triangle, then the triangles are congruent. |

Side-Side-Side Similarity Theorem | If three or more triangles have three corresponding, proportional sides, then the triangles are similar. |

Square Root Property of Equality | For any real number a, the square root of a squared equals a. |

Substitution Property of Equality | For real numbers a, b, and c if a equal b, then a minus c equals b minus c. |

Supplementary Angles of a Trapezoid Theorem | Consective angles between the two bases of a trapezoid are supplementary. |

Symmetric Property of Equality | For real numbers a and b, if a equals b, then b equals a. |

Transitive Property of Equality | For real numbers a, b, and c, if a equals b and b equals c, then a equals c. |

Triangle Exterior Angle Theorem | The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. |

Triangle Inequality Theorem | The sum of the two lengths of any two sides of a triangle is greater than the third side. |

Corollary to the Triangle Inequality Theorem | The length of the third side of the triangle is less than the total and greater the absolute value of the difference of the other two sides. |

Triangle Proportionally Theorem | If a line is parallel to one side of a triangle and also intersects the two sides, the line divides the sides proportionally. |

Triangle Sum Theorem | The sum of the measures of the angles in the triangle is 180 degrees. |

Vertical Angles Theorem | Vertical angles are congruent |