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Transcript
Chapter 1- Study Guide Geometry- Mr. Barnes Vocabulary 1-1 Patterns and Inductive Reasoning Inductive Reasoning-(p.4) is a type of reasoning in which all possibilities are considered and then the unwanted ones are proved false. The remaining possibilities must be true. Ex. Eduardo spent more than $60 on two books at the store. Prove that at least one book costs more than $30 Proof: Suppose neither costs more than $30. Then he spent no more than $60 at the store. Since this contradicts the given information, at least one book costs more than $30. In Math: Find a pattern (change) for each sequence. Use the pattern to show the next two terms in the sequence. 4, 8, 16, 32 x2 x2 x2 Each term is twice the preceding term. The next two terms are: 2x32=64 and 2x64=128 Hint: Find the “change” between terms. Sometimes it’s addition, sometimes its multiplication, sometimes it is something all together different Coordinate (p.5) A coordinate is a point defined on a line segment. Conjecture (p.5) a conjecture is a conclusion reached by using inductive reasoning. Ex. As you walk down the street, you see many people holding unopened umbrellas. You conjecture that the weather forecast must call for rain. In Math: Consider one more example. Suppose I start with the number 60 and try to divide 60 by a few numbers smaller that 60. ex. I try to divide 60 by 1,2,3,4,5 etc. I observe that 60 is divisible by each of these numbers. So I might be tempted to make a conjecture that every number less than 60 is a divisor of 60. To check the validity of my conjecture I may try a few more numbers less than 60 and see whether they divide 60. ex. I may try 6, 10, 12, 15, 20 etc. as divisors and say that my conjecture seems to be true. But if you want to shoot down my conjecture all you need to do is to point a single exception to the rule I have guessed. For example you may point out that 7 is less than 60 but 60 is not divisible by 7. Hence my conjecture has been demonstrated to be wrong. The value of 7 is called a 'counter example' in this context. Thus we observe: Every conjecture may not be true. A conjecture can be refuted by giving a counter example. Almost all mathematical results, before they are proved; start their life as conjectures. Some conjectures remain unproved for many many years. So unless and until a conjecture is proved to be true or a counter example is found it remains a conjecture. Only when a conjecture has been proved to be true it becomes a mathematical result (theorem) Counter Examples (p.17) a counter example to a statement is a particular example or instance of the statement that makes the statement false Ex. 'The opposite of a number is always positive.' Counterexample for the above statement: The opposite of 2 is - 2, a negative number. 1-2 Drawing, Nets and Other Models Isometric Drawing (p.10) An isometric drawing of a three dimensional object shows a corner view of a figure. It is not drawn in perspective and distances are not distorted. Ex. Make an isometric drawing of the cube structure at the left. Orthographic Drawing (p.11) another way to show a three dimensional figure. It shows the top view, front view and right-side view. Ex. Make an orthographic drawing from the isometric drawing at the left. Foundation Drawing (p.11) a foundation drawing shows the base structure and the height of each part. Ex. Net (p.12) a net is a two dimensional pattern that you can fold to form a three dimensional figure Ex. Identifying a Net The pattern is a net because you can fold it to form a cube. A and C, B and D, and E and F are on opposite faces. Ex. Drawing a Net Packaging Draw a net for the graham cracker box. Label the net with its dimensions. 1-3 Points, Lines and Planes Point (p.17) A point is a location. A point has no size. It is represented by a small dot and is named by a capital letter. Space (p.17) the set of all points Line (p.17) A line is a series of points that extends into two opposite directions without end. You can name a line by any two points on the line, such as AB or line AB. Another way to name a line is with a single lowercase letter, such as line t. Ex. Collinear Points (p.17) Collinear points lie on the same line Ex. Are points E, F, and C collinear? If so, name the line on which they lie. Points E, F, and C are collinear. They lie on line m. Are points E, F, and D collinear? If so, name the line on which they lie. Points E, F, and D are not collinear. Plane (p.17) A plane is undefined. You can think of a plane as a flat surface that has no thickness. A plane contains many lines and extends without end in the direction of its lines Ex. Postulate (p.18) A postulate, or axiom, is an accepted statement or fact. Postulate 1-1 Through any two points there is exactly one line. Line t is the only line that passes through points A and B. Postulate 1-2 If two lines intersect, then they intersect in exactly one point. and intersect at C. Postulate 1-3 If two planes intersect, then they intersect in exactly one line. Plane RST and plane STW intersect in Postulate 1-4 Through any three noncollinear points there is exactly one plane. 1-4 Segments, Rays, Parallel Lines and Planes Segment (p.23) a segment is the part of the line consisting of two points, called endpoints, and all points between. Ex. Ray (p.23) A ray is the part of a line consisting of one endpoint and all of the points of the line on one side of the endpoint Ex. Opposite rays (p. 23) Opposite Rays are two collinear rays with the same endpoint. Opposite rays always form a line. Ex. Parallel Lines (p.24) Two lines are parallel if they lie in the same plane and do not intersect. The symbol means “is parallel to” Ex. Segments or rays are parallel if they lie in parallel lines. They are skew if they lie in skew lines. and are skew because and are skew. Skew lines (p.24) Skew Lines are noncoplanar; therefore, they are not parallel and do not intersect. Parallel Planes (p. 24) are planes that do not intersect. 1-5 Measuring Segments and Angles Coordinate of a point (p.25,43)The coordinate of a point is its distance and direction from the origin of a number line. The coordinates of a point on a coordinate plane are in the form (x, y), where x is the x-coordinate and y is the y-coordinate. Congruent Segments (p. 31) Two segments with the same length. The congruence symbol ( ) shows that two figures are equal (=) in size and similar (~) in shape. Ex. Two segments with the same length are congruent ( ) segments. In other words, if AB = CD, then these statements interchangeably. . You can use As illustrated above, segments can be marked alike to show they are congruent. Midpoint (p.32) a midpoint of a segment is the point that divides the segment into two congruent segments. A midpoint, or any line, ray, or other segment through a midpoint, is said to bisect the segment. Ex. 1-6 Measuring Angles (p.36) Angle (p.36) An angle( ) is formed by two rays with the same endpoint. The rays are the sides of the angle and the common endpoint is the vertex of the angle One way to measure an angle is in degrees. To indicate the size or degree measure of an angle, write a lowercase m in front of the angle symbol. The degree measure of angle A is 80. You show this by writing m A = 80. Acute Angle (p.37) an acute angle is an angle whose measure is between 0-90 Ex. Congruent Angles (p.37) congruent angles are angles that have the same measure Obtruse Angle (p.37) an obtruse angle is an angle whose measure is between 90 and 180 Ex. Right Angle(p. 37) A right angle is an angle whose measure is 90. Ex. Straight Angle (p. 37) A Straight Angle is an angle whose measure is 180. Ex. Construction (p. 44) A construction is a geometric figure made with only a straightedge and a compass Perpendicular Lines (p.45) Perpendicular lines that intersect and form right angles. The symbol means is “perpendicular to” Angle Bisector (p. 46) an angle bisector is a ray that divides an angle into congruent angles CHAPTER 1 FORMULAS Find the Length of a Segment Postulate 1-5- Ruler Postulate (p.31) The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers. Ex. The distance between points C and D on the ruler is 3. You can use the Ruler Postulate to find the distance between points on a number line. Comparing Segment Lengths (p. 32) Find AB and BC. AB = | −8 − (−5) | = | −3 | = 3 BC = | −5 − (−2) | = | −3 | = 3 AB = BC or Segment Addition Postulate Postulate 1-6 If three points, A, B and C are collinear and B is between A and C, then AB+BC+AC Ex. Algebra If DT = 60, find the value of x. Then find DS and ST. Finding the Midpoint (p.33) A midpoint of a segment is a point that divides a segment into two congruent segments. A midpoint, or any line, ray, or other segment through a midpoint, is said to bisect the segment. Ex. Finding Lengths Algebra C is the midpoint of . Find AC, CB, and AB. AC and CB are both 11, which is half of 22, the length of . Using the Angle Addition Postulate 1-8 (p.38) What is m TSW if m RST = 50 and m RSW = 125? Using the Distance Formula