1.1 Getting Started A 1 F m point location in space. E line The {set} of infinite points arranged in a straight figure that extends infinitely far in both directions. which each point is assigned a numerical value. line segment (segment) part of a line with two endpoints. ray part of a line that begins at one endpoint and extends infinitely far in only one direction. angle figure made up of two rays (sides) with a common endpoint (vertex). vertex side - triangle a figure with three sides and three angles. intersection - subset of elements that are in both sets. union - subset of elements that are in one set or the other set or both sets. EXAMPLES: 1) Given: A = {-3, -4, 0, 5, 10} 2) 3) raw a diagram s.t. = {-4, -5, -7, 0, 1, 5, 3} A A = A A = {2, 4, 6} Find: A = Find: A A = A = A = A = 4) Find A A & A A 5) Find the union and intersection of prime numbers less than 20 and whole numbers between 0 and 11. #4 is wrong on video. Make sure you understand why?
1.2 Measurement of Segments and Angles measurement of a segment A = 7.2 in measurement of an angle - m XYZ = 32 G O protractor Tool used to measure angles. U S A degree - unit of measurement for an angle. acute - 0 < m < 90 obtuse - 90 < m < 180 right - m = 90 straight - m = 180 ongruent segments/angles segments/angles that have the same measurement. A = A m Q = m R Q R E tick marks marks used to indicate angles and segments. Parts of a degree -> minutes and seconds 1 = 60 A 1 = 60 Thus, 87.5 = 87 30, 60.4 = 60 24, 90 = 89 60 or 89 59 60 F EXAMPLES: 1) hange 41.6 to minutes/seconds and change 60 42 30 to degrees 2) 22 37 9 90 3) Find the restrictions on x + 5 42-76 23 52 if is acute (2x+40) 4) 4:00 5) 10:00 6) 5:30 7) 9:56 8) 7:06
1.3 ollinearity, etweenness, and Assumptions ollinear (points) points that lie on the same line. A noncollinear (points) points that do not lie on the same line. betweenness of points A point is between two endpoints of a segment if all three points are collinear and none of the points are the same. triangle inequality The sum of any two sides of a triangle is greater than the third side. Assumptions from a diagram I may assume: I may not assume: 1) Straight lines and angles 1) Right angles 2) ollinearity of points 2) segments 3) etweeness of points 3) angles 4) Relative positions of points 4) Relative sizes of segments and angles EXAMPLES: 1) Given: Two sides of a triangle are 7 and 11. Find all possible side lengths for the third side. 2) Find m AT 3x+20 2x+10 M A T
1.4 eginning Proofs Angle Addition Postulate vs. Segment addition postulate two column proof A method of proof that provides statements and their reasons in organized columns. theorem A math statement that has been proven true. procedure: 1) Present a theorem 2) Prove the theorem 3) Understand and memorize the theorem 4) Use the theorem Acceptable Reasons for Proofs: 1) 2) 3) 4) 5) Thm 1 - Thm 2 Examples: Proof of Thm 1 Proof of Thm 2 Given: A is Given: A is straight is EF is straight Prove: A Prove: A EF
1.5 ivision of Segments/Angles bisect bi -> 2 sect -> short for section -> verb: To cut into 2 equal sections. segment bisector A line or part of line that cuts a segment in half. midpoint The bisection point of a segment. trisect tri -> 3 sect -> short for section -> verb: To cut into 3 equal sections. segment trisector 2 lines or parts of lines that divide a segment into 3 parts. trisection points The 2 points at which is segment is trisected. angle bisector A ray that divides an angle into 2 angles. angle trisectors Two rays that divide an angle into 3 angles. Examples: 1) raw A with a measure of 87 13 and angle bisector Find m A 2) Given: M bisects A 3) Given: H HF Prove: AM M (AM = M) A M Prove: H is midpoint of F G H E F
1.6 Paragraph Proofs paragraph proofs A method of proof that uses a paragraph format. counterexample An example that can be used to prove a statement is false. Like a good English paper, a good paragraph proof has an introduction, a body, and a conclusion. intro -> body -> conclusion -> Examples: We will talk through proofs 7 and 8 on p. 38 No need to copy these down, we will be doing mainly two-column proofs.
1.7 eductive Structure eductive structure/reasoning A systematic method of coming up with a conclusion by using previously assumed or proven statements. Elements of the deductive structure: 1) Undefined terms 2) Postulates 3) efinitions 4) Theorems, etc. postulate A statement that can be assumed true without proof. What is an important characteristic of a EFINITION? versible\ Is this true for theorems/postulates? ssarily conditional statement (implication) A declarative statement in if then form. eclarative Sentence All right angles are congruent. onditional Form If 2 angles are right, then they are congruent. hypothesis Part of a conditional statement that follows If conclusion Part of conditional statement that follows then converse A statement in which the hypothesis and conclusion are switch q -> If q, then p biconditional statement A statement in which the conditional and the converse are both true (if and only if). Examples: 1) eclarative Statement -> All straight angles are congruent. onditional statemtent -> hypothesis -> conclusion -> converse -> Is this a biconditional statement? 2) eclarative Statement -> All right angles measure 90. onditional statemtent -> hypothesis -> conclusion -> converse -> Is this a biconditional statement?
1.8 Statements of Logic negation (~) noting a statement -> negating a neg. statement will make it pos. inverse A statement formed by negating both the hypothesis and conclusion of a conditional statement. If p, then q -> If ~p, then ~q contrapositive A statement formed by taking the inverse of the converse of a conditional statement. If p, then q -> If ~q, then ~p venn diagram A tool used to determine if a conditional statement is true or false. Thm 3 If a conditional statement is true, then the contra-positive will be true. If p, then q If ~q, then ~p chain rule Law of syllogism -> If p q and q r then p r. Examples: (TRANSITIVE PROPERTY) 1) If you go to irmingham Groves HS, then you live in Michigan. (T or F) onverse (T or F) -> Is it a biconditional? Inverse (T or F) -> ontra-postive (T or )-> Venn-iagram 2) If points are collinear, then they lie on the same line. (T or F) onverse (T or F) -> Is it a biconditional? Inverse (T or F) -> ontra-postive (T or )-> Venn-iagram
1.9 Probability probability The chance that something will or will not happen. 0 1 or 0% - 100% Steps to determine probability 1) etermine the total possible number of outcomes. 2) etermine the number of favorable/desired outcomes. P(E) = () Examples: 1) If three of the points are picked at random, what is the probability that they will be collinear? A E 2) Given 6 angles with the following measurements, find the probability that if you select two of them at random that they will be complementary (add up to 90). 20, 30, 40, 50, 60, 70 3) A point is chosen at random on PT, what is the probability that it is within 4 units of R. P R T -3 2 11