Reasoning in Geometry 1.1 Patterns and Inductive Reasoning 1.4 Conditional Statements and Their Converses 1.3 Postulates 1.2 Points, Lines, and Planes 1.6 A Plan for Problem Solving 1.5 Tools of the Trade
2) Solve the equation. Check your answer.5 Minute-Check1) Both answers can be calculated. Which one is right? What makes it right? What makes the other one incorrect?3) If a dart is thrown at the circle to the right, what is the probability that it will land in a yellow sector? The odds?
Find the value or values of the variable that makes each equation true.220.127.116.11.5.6. Find the next three terms of the sequence. 6, 12, 24, . . . g = 21x = 5y = 4 or y = - 4z = - 2 648, 96, 1925 Minute-Check
Patterns and Inductive ReasoningIf you were to see dark, towering clouds approaching, you might want to take cover. Your past experience tells you that a thunderstorm is likely to happen.When you make a conclusion based on a pattern of examples or past events, you are using inductive reasoning.You will learn to identify patterns and use inductive reasoning.
Patterns and Inductive ReasoningYou can use inductive reasoning to find the next terms in a sequence.Find the next three terms of the sequence:3, 6, 12,24, 48, 96,Find the next three terms of the sequence:7, 8, 11,16,3223,
Patterns and Inductive ReasoningDraw the next figure in the pattern.
Patterns and Inductive ReasoningA _________ is a conclusion that you reach based on inductive reasoning.In the following activity, you will make a conjecture about rectangles.conjecture1) Draw several rectangles on your grid paper.2) Draw the diagonals by connecting each corner with its opposite corner. Then measure the diagonals of each rectangle.3) Record your data in a tableMake a conjecture about the diagonals of a rectangle
Diagonal 1Diagonal 2Rectangle 17.5 inches7.5 inches
Patterns and Inductive ReasoningA conjecture is an educated guess.Sometimes it may be true, and other times it may be false.How do you know whether a conjecture is true or false?Try different examples to test the conjecture.If you find one example that does not follow the conjecture, then the conjecture is false.Such a false example is called a _____________.counterexampleConjecture: The sum of two numbers is always greater than either number.Is the conjecture TRUE or FALSE ?Counterexample: -5 + 3 = - 2- 2 is not greater than 3.
Patterns and Inductive Reasoning
5 Minute-CheckFind the next three terms of each sequence.18.104.22.168.59, 63, 6715.5, 20.5, 26.5 -2 + 4 = 2 and 2 < 4Draw the next figure in the pattern shown below.Find a counterexample for this statement:The sum of two numbers is always greater than either addend.5) If a dart is thrown at the circle to the right, what is the probability that it will land in a shaded sector? The odds?
Points, Lines, and PlanesGeometry is the study of points, lines, and planes and their relationships. Everything we see contains elements of geometry.You will learn to identify and draw models of points, lines, and planes, and determine their characteristics.
Points, Lines, and PlanesA ____ is the basic unit of geometry.pointPOINT: A point has no ____.size Points are named using capital letters. The points at the right are named point A and point B.
Points, Lines, and PlanesA ____is a series of points that extends without end in two directions.lineLINE: A line is made up of an ______ _______ of points.infinite number The ______ show that the line extends without end in both directions. A line can be named with a single lowercase script letter or by two points on the line.arrows The line below is named line AB, line BA, or line l.
Points, Lines, and Planes1) Name two points on line m. Possible answers: point R and point S point R and point T point S and point T2) Give three names for the line.Possible answers:NOTE: Any two points on the line or the script letter can be used to name it.
Points, Lines, and PlanesThree points may lie on the same line. These points are _______ .collinearPoints that DO NOT lie on the same line are __________ .noncollinear points R, S, and point T points U, S, and point V
Points, Lines, and PlanesThree points may lie on the same line. These points are _______ .collinearPoints that DO NOT lie on the same line are __________ .noncollinear points R, S, and point V points R, S, and point U points S, T, and point V points R, V, and point U points R, T, and point V points R, T, and point U
Points, Lines, and PlanesA ___ has a definite starting point and extends without end in one direction.rayendpoint A ray is named using the endpoint first, then another point on the ray. The ray above is named ray AB.Rays and line segments are parts of lines.
Points, Lines, and PlanesLINE SEGMENT: A line segment is part of a line containing two endpoints and all points between them. A line segment is named using its endpoints. The line segment above is named segment AB or segment BA.Rays and line segments are parts of lines.A ___________ has a definite beginning and end.line segment
Points, Lines, and Planes1) Name two segments.ABUDPossible Answers:2) Name a ray.Possible Answers:
Points, Lines, and PlanesA _____ is a flat surface that extends without end in all directions.planePoints that lie in the same plane are ________.Points that do not lie in the same plane are ___________.coplanarnoncoplanarPLANE: For any three noncollinear points, there is only one plane that contains all three points. A plane can be named with a single uppercase script letter or by three noncollinear points. The plane at the right is named plane ABC or plane M.BA MC
Hands On Place points A, B, C, D, & E on a piece of paper as shown. Fold the paper so that point A is on the crease. Open the paper slightly. The two sections of the paper represent different planes.1) Name three points that are coplanar. ______________________2) Name three points that are noncoplanar. ______________________3) Name a point that is in both planes. ______________________Answers (may be others)A, B, & CD, A, & BA
Points, Lines, and Planes
5-Minute Check1) Name three points on line r2) Give three other names for line r3) Name two segments that have point F as an endpoint.4) Name three different rays.5) Are points C, E, and F collinear or noncollinear?D, E, Fnoncollinear
PostulatesYou will learn to identify and use basic postulates about points, lines, and planes.
PostulatesGeometry is built on statements called _________.Postulates are statements in geometry that are accepted to be true.postulatesPostulate 1-1: Two points determine a unique ___.There is only one line that contains Points P and QPostulate 1-2: If two distinct lines intersect, then their intersection is a ____. Lines l and m intersect at point T Postulate 1-3: Three noncollinear points determine a unique _____. There is only one plane that contains points A, B, and C.linepointplane
PostulatesABCPoints A, B, and C are noncollinear. 1) Name all of the different lines that can be drawn through these points. Point C
Postulates1) Name all of the planes that are represented in the figure.There is only one plane that contains three noncollinear points.plane ABC (side)plane ACD (side)plane ABD (back side)plane BCD (bottom)
PostulatesPostulate 1-4: If two distinct planes intersect, then their intersection is a ___.Plane M and plane N intersect in line DE.line
PostulatesName the intersection of plane CDG and plane BCD.planes ADF and CDF
5-Minute Check1) At which point or points do three planes intersect?2) Name the intersection of plane ABC and plane ACD.3) Are there two planes in the figure that do not intersect?At each of the points A, B, C, and D.NoPlanes ABD and BCD.One, (Point B)
Conditional Statements and Their ConversesIn mathematics, you will come across many _______________.For Example:If a number is even, then it is divisible by two.If then statements join two statements based on a condition:A number is divisible by two only if the number is even.Therefore, if then statements are also called __________ __________ .if-then statementsconditional statementsYou will learn to write statements in if-then form and write the converse of the statements.
Conditional Statements and Their ConversesConditional statements have two parts.The part following if is the _________ .hypothesisThe part following then is the _________ .conclusionIf a number is even, then the number is divisible by two.a number is eventhe number is divisible by two.Hypothesis: Conclusion:
Conditional Statements and Their ConversesHow do you determine whether a conditional statement is true or false?If it is the 4th of July (in the U.S.), then it is a holiday.TrueThe statement is true because the conclusion follows from the hypothesis.If an animal lives in the water, then it is a fish.FalseYou can show that the statement is false by giving one counterexample.Whales live in water, but whales are mammals, not fish.
Conditional StatementTrue or FalseWhy?