Geometry
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Addition Properties
| Question (memorize) | Answer (memorize) |
|---|---|
| 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 (memorize) | Answer (memorize) |
|---|---|
| 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 (memorize) | Answer (memorize) |
|---|---|
| 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 ac < bc; if a < b and c < 0, then bc < ac |
Proved Properties (Theorems)
| Question (memorize) | Answer (memorize) |
|---|---|
| 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 < <sup>b/c; if a < b, and c < 0, <br> 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 (memorize) | Answer (memorize) |
|---|---|
| P1 | 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). |
| P2 | Through any two different points, there is exactly one line. |
| P3 | Through any three points (which are not on one line), there is exactly one plane. |
| P4 | If two points lie in a plane, then the line containing them lies in that plane. |
| P5 | If two different planes intersect, then their intersection is a line. |
| P6 | Between any 2 points there is a unique distance. |
| P7 | (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 (memorize) | Answer (memorize) |
|---|---|
| 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 | a2 + b2 = c2 in a right triangle when a and b are the legs and c is the hypotenuse. |
See Geometry 2 and Math for more.
