# Geometry

rename## Addition Properties

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

closure | a + b is unique real number |

commutative | a + b = b + a |

associative | (a + b) + c = a + (b + c) |

additive of zero | a + 0 = a and 0 + a = a |

additive inverses | a + (-a) = 0 and (-a) + a = 0 |

## Multiplication Properties

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

closure | a * b is a unique real number |

commutative | ab = ba |

associative | (ab)c = a(bc) |

multiplication postulate of one | a * 1 = a and 1 * a = a |

multiplicative inverses | a * ^{1}/_{a} = 1 |

distributive | a(b + c) = ab + ac and (b + c)a = ba + ca |

## Equality and Inequality

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

reflexive | a = a |

symmetric | if a = b, b = a |

transitive | if a = b and b = c, then a = c |

comparison | only one is true: a < b, a = b, b < a |

transitive postulate | if a < b and b < c, then a < c |

additive postulate | if a < b, then a + c < b + c |

multiplicative postulate | if a < b and 0 < c,then a c < bc; if a < b and c < 0,then b c < ac |

## Proved Properties (Theorems)

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

Addition (equality) | if a = b, then a + c = b + c and c + a = c + b |

Subtraction (equality) | if a = b, then a - c = b - c and c - a = c - b |

Multiplication (equality) | if a = b then ac = bc |

Division (equality) | if a = b, and c ≠ 0, then ^{a}/_{c} = ^{b}/_{c} |

Subtraction (inequality) | if a < b, then a - c < b - c and c - a > c - b |

Division (inequality) | if a < b, and c > 0, then ^{a}/_{c} < ^{b}/_{c}; if a < b, and c < 0, then ^{a}/_{c} > ^{b}/_{c} |

Substitution Principle | if a = b, a can be replaced by b |

Zero-Product Property | if ab = 0, then a = 0 or b = 0 |

## Postulates

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

P_{1} | A line contains at least two points. A plane contains at least three points (not all on one line). Space contains at least 4 points (not all in one plane). |

P_{2} | Through any two different points, there is exactly one line. |

P_{3} | Through any three points (which are not on one line), there is exactly one plane. |

P_{4} | If two points lie in a plane, then the line containing them lies in that plane. |

P_{5} | If two different planes intersect, then their intersection is a line. |

P_{6} | Between any 2 points there is a unique distance. |

P_{7} | (Ruler Postulate) AB= lx-yl and there is a one-to-one correspondence with all real numbers and points on the number line. |

## Theorems

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

Theorem 3-1 | If 2 lines intersect, they intersect at exactly one point. |

Thoerem 3-2 | If a point lies outside a line, exactly one plane contains the point and the line. |

Thoerem 3-3 | If 2 lines intersect, only one plane contains both lines. |

Thoerem 3-4 | On a ray there is exactly one point at a given distance from the ray's endpoint. |

Thoerem 3-5 | A segment has exactly one mid-point. |

Pythagorean Theorem | a^{2} + b^{2} = c^{2} in a right triangle when a and b are the legs and c is the hypotenuse. |

See Geometry 2 and Math for more.