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AMDG NAME ______________________________ Teacher: ____________________________ Class and Period:______________________ Date: _______________________________ Introduction to Geometry Axiomatic System Notebook (Ch 1ASN) Ch. Sec. 1.1 Term, Example, Proof A _____________ is a place in space. It has no length, width, or height. We use capital letters to name points. Two points determine a _______________. All __________ are straight. It has infinite length (but no width of height). It has ____ dimension. We name a line by a lower case script letter, or by any two points on the line. A ________________________________ is a part of a line. It has finite length. It is named by the endpoints. A ___ has one endpoint, and infinite length (in the direction opposite the endpoint). The endpoint is always noted first in the name. An ___________ is formed by two rays that share the endpoint. The shared endpoint is called the _____________. The rays are called the ______. Names of angles should include 3 points with the vertex noted as the MIDDLE letter. Sometimes angles are numbered. 1 AMDG A _________________ has three segments as its sides. It also has three angles. It is named by the three points that are the vertices. A _________________ has length and width (that is it is 2 dimensional), and is defined by 3 points. 2 AMDG A triangle is the _____ () of three segments. Union implies () OR, which is always INCLUSIVE in mathematics. Another example: Meet me on 35th OR Wisconsin: (35th St.) (Wisconsin Av.) An ____________________ () of two sides of a triangle is the vertex. Intersection AND, which is exclusive. It’s the overlap. Another example: Meet me on 35th AND Wisconsin: (35th St.) (Wisconsin Av.) 3 AMDG 1.2 Angles are measured in degrees (), minutes (‘), and seconds (“). There are ____ minutes in a degree, and _____ seconds in a minute. An _____________ angle is an angle whose measure is greater than 0 and less than 90 degrees. A _____________ angle is an angle that measures 90 degrees. An __________ angle is an angle whose measure is greater than 90 and less than 180 degrees. A __________straight angle is an angle whose measure is 180 degrees. A straight angle is also a line. Def ____________________________ are angles that have the same measure. Def _____________________________ are segments that have the same length. ___________________________ are used to indicate congruent angles or segments. 1.3 Def Points that lie on the same line are __________________________. Points that do not lie on the same line are _______________________. 4 AMDG Betweenness of points: In order for us to say that a point is ____________ two other points, all three points must be _________________. Triangle Inequality: The sum of two sides is always (greater than/equal to/ less than) the third side. Interpreting a Diagram: You should Assume Straight lines and angles Collinearity of points Betweenness of points Relative position of points 1.7 You should NOT assume Right angles Congruent segments Congruent angles Relative sizes of segments and angles A ___________________________ is an axiomatic system that contains 4 elements: A ________________ is an unproven assumption. A postulate is NOT always reversible. A _________________ states the meaning of a term or idea. A definition is always reversible. Points lie on the same line collinear Collinear points lie on the same line A _______________ is a mathematical statement that can (and should) be proven. Theorems are NOT always reversible. Poodle dog Dog poodle 5 AMDG 1.8 If you are not willing to learn, no one can help you. If you are determined to learn, no one can stop you. Conditional Statement Negation Converse Inverse Contrapositive Truth tables get a little more complicated when conjunctions and disjunctions of statements are included. ^ is INTERSECTION, v is UNION ^ is AND, v is OR This is a brief introduction to Boolean Algebra, and is the logic used in search engines. 6 The truth table for an implication, or conditional statement looks like this: AMDG 1.4 Purposes of proof Mnemonic: Very Easy Class Don’t Cheat VERIFY the truth of a mathematical statement. EXPLAIN why the statement is true. COMMUNICATE mathematical knowledge. DISCOVER new math (that is axioms) CREATE an axiomatic system (or a system of truths) A proof is the logical process, not the format of the argument. A _________________________________________ is one format of a proof. Thm Thm 7 AMDG 1.5 Def A point (or segment, ray, or line) that divides a segment into two congruent segments ___________________ the segment. The bisection point is called the ________________ of the segment. Def Two points (or segments, rays, or lines) that divide a segment into three congruent segments ___________________ the segmetn. The two points at thich the segment is divided are called the ______________________ of the segment. Def A ray that divides an angle into two congruent angles ______________ the angle. The dividing ray is called the ________________________. Def Two rays that divide an angle into three congruent angles __________________ the angle. The dividing rays are called _____________________________. 1.9 Two Basic Steps for Probability Problems 1. Determine all possiblities in a logical manner. Count them. 2. Detemine the number of these possibiliites that are “favorable.” We shall call these winners. number of winners Probability = number of possibilities 8