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Chapter 1: 1-1: (page 1) Points, Lines, Planes, and Angles A Game and Some Geometry (page 1) In the figure below, you see five points: A,B,C,D, and E. Use a centimeter ruler to find the requested distances. Please measure to the nearest tenth of a centimeter. C A B D E 1. The distance from A to C is cm and the distance from B to C is cm. 2. The distance from A to D is cm and the distance from B to D is cm. 3. Both points C and D are said to be they are equally distant from A and B. from A and B because 4. Another point on the diagram that is equidistant from A and B is point . 5. The number of points that are equidistant from points A and B are . 6. The geometric figure that would be formed by all these points is a . Locate a point 9 cm from point A and call it point F. Locate all the points that are 9 cm from point A. 7. The number of points that are 9 cm from point A is . 8. The geometric figure that would be formed 9 cm from point A is a _______________. Locate a point that is 9 cm from points A & B and call it point G. Locate all the points that are 9 cm from points A & B. 9. The number of points that are 9 cm from points A and B is ___________________. 10. The geometric figure that would be formed by connecting these points is a Assignment: Written Exercises, pages 3 & 4: 1 to 10 . 1-2: Points, Lines, and Planes (page 5) What real life objects are made up of points (dots)? Look up the definitions for Point, Line, and Plane in the glossary of your textbook. POINT: LINE: PLANE: Point, Line, and Plane are the basis to defining other geometry terms, they are accepted without , therefore they are considered terms. Descriptions of the Three (3) Undefined Terms picture symbol description -------------------------------------------------------------------------------------------------------------------POINT * has length, width, or thickness * has dimensions * occupies space -------------------------------------------------------------------------------------------------------------------LINE * has , but width or thickness * has an set of points that extends in directions -------------------------------------------------------------------------------------------------------------------PLANE * has and but thickness * has an set of points that extends in * directions surface SPACE: the set of points. GEOMETRIC FIGURE: a of points. [figure 1] A [figure 2] B V C W X Y D Z COLLINEAR POINTS: points that lie on the same . example from figure 1: NONCOLLINEAR POINTS: points that do lie on the same line. example from figure 1: COPLANAR POINTS: points that lie in the same . example from figure 2: NONCOPLANAR POINTS: points that do example from figure 2: lie in the same plane. INTERSECTION (of two or more figures): the set of to all figures. examples: (1) Lines (2) Planes Assignment: Written Exercises, pages 7 to 9: 1 to 35 odd #’s Prepare for Quiz on Lessons 1-1 & 1-2 that are common Segments, Rays, and Distance 1-3: Point B is (page 11) A and C if it lies on line AC ( A B ). C SEGMENT: consists of points A and C and all points that are example: A A and C. C RAY: consists of AC and all other points P such that C is example: A C A and P. P OPPOSITE RAYS: two rays that have the same example: X O and form a line. Y Geometry and Algebra are brought together with the A | -4 B | -3 C | -2 D | -1 Every point is paired with a E | 0 line. F | 1 G | 2 H | 3 I | 4 J | 5 . Every number is paired with a . 1 - to - 1 Correspondence A corresponds to or A ! _______, B corresponds to or B ! _______ , etc. LENGTH (of a segment): the AB means the between its endpoints. of AB or the between points A & B. AB = a - b = b - a examples: CH = ______ AJ = ______ BF = ______ POSTULATES (Axioms): statement accepted without POSTULATE 1 . RULER POSTULATE (1) The points on a line can be paired with the real numbers in such a way that any two points can have coordinates and . (coordinatized line) (2) Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their . (distance) POSTULATE 2 SEGMENT ADDITION POSTULATE If B is between A and C, then: AB + BC = . example: Given the diagram with AB = 2x, BC = x-1, and AC = 23, find AB and BC. A B C AB = BC = CONGRUENT: objects that have the same _____________ and _____________. CONGRUENT SEGMENTS: segments that have equal _____________. example: A C AB CD ! AB ____ CD B D _______ expresses a relationship between numbers. _______ expresses a relationship between geometric figures, NOT numbers. MIDPOINT of a SEGMENT: the point that divides the segment into_______ congruent segments. example: A M B AM ____ MB, then AM ____ MB ! M is the _____________ of AB. BISECTOR of a SEGMENT: a _____________, _____________, _____________, or plane that intersects the segment at its midpoint. Assignment: Written Exercises, pages 15 & 16: 1 to 45 odd #’s and 46, 47 Angles (page 17) ANGLE: a figure formed by 2 rays that have the same . 1-4: example: Naming an angle may be done as follows: . To measure angles (in degrees in this course), use a . Classification of Angles Acute Angle: measure is between and Right Angle: measure equals degrees. degrees. Obtuse Angle: measure is between and Straight Angle: measure equals degrees. degrees. Angle in a Plane (exclude straight angle) Plane is separated into 3 parts: (1) the angle itself (2) the interior of the angle (3) the exterior of the angle How many angles are shown in the diagram below? Name the different angles: X 2 3 O Y Z PROTRACTOR POSTULATE POSTULATE 3 On line AB in a given plane, choose any point between A and B. Consider ray OA and ray OB and all rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: (a) ray OA is paired with , and ray OB with . (b) If ray OP is paired with x, and ray OQ with y, then Q A = x!y . P O B ANGLE ADDITION POSTULATE POSTULATE 4 If point B lies in the interior of !AOC , then: + = A . B O B C A O C If !AOC is a straight angle and B is any point not on line AC, then: + = 180º . CONGRUENT ANGLES: angles that have measure. example: m!A _____ m!B A B !"A _____ "B ADJACENT ANGLES: two angles in a that have a common vertex and a common side, but no common interior points. (adj. !' s) examples: Are ! 1 and ! 2 adjacent angles? Circle Yes or No. (1) (2) (3) 1 1 2 1 Yes or No 2 2 Yes or No (4) (5) Yes or No (6) 1 2 2 1 1 2 Yes or No Yes or No Yes or No BISECTOR of an ANGLE: the ray that divides the angle into 2 congruent m!XOY _____ m!YOZ X example: then !XOY _____ !YOZ O ! OY __________ "XOZ Z angles. MORE EXAMPLES: In diagram, OB bisects !AOC. A 2 3 D (1) (2) B 1 O m! 1 = 2x + 5 and m! 2 = 3x -12. C m! 1 = ________ m! 2 = ________ m! 3 = ________ m! 1 = x + 4 and m! 3 = 2x + 8. m! 1 = ________ m! 2 = ________ m! 3 = ________ Assignment: Written Exercises, pages 21 & 22: 1 to 35 odd #’s, 24 Prepare for Quiz on Lessons 1-3 & 1-4 1-5: Postulates and Theorems Relating Points, Lines, & Planes The phrases, “exactly one” and “one and only one” imples POSTULATE 5 A line contains at least A plane contains at least Space contains at least and . points; points not all in one line; points not all in one plane. POSTULATE 6 Through any two points there is exactly POSTULATE 7 Through any three points there is at least line. plane; Through any three noncollinear points there is exactly plane. POSTULATE 8 If points are in a plane, then the line that contains the points is in that plane. POSTULATE 9 If two planes intersect, then their intersection is a THEOREMS: statements that can be . . NOTE: Writing proofs will be covered in Chapter 2. THEOREM 1-1 If two lines intersect, then they intersect in exactly point. NO PROOF - example: THEOREM 1-2 Through a line and a point not in the line there is exactly plane. NO PROOF - example: THEOREM 1-3 If two lines intersect, then exactly NO PROOF - example: (page 22) plane contains the lines. Relationships between points: Two points be collinear. Three points be collinear or noncollinear. Three points be coplanar. Three noncollinear points determine a Four points . be coplanar or noncoplanar. Four noncoplanar points determine Space contains at least . noncoplanar points. Three ways to determine a plane: (1) _________________________ noncollinear points determine a plane, ex: (2) A line and a point not on the determine a plane, ex: (3) _________________________ intersecting lines determine a plane, ex: Relationships between two lines in the same plane: Two lines are either parallel or they in exactly one point. examples: Relationships between a line and a plane: A line and a plane are either parallel, or they the plane the line. in exactly one point, or examples: Relationships between two planes: Two planes are either parallel or they in a line. examples: Assignment: Written Exercises, pages 25 & 26: 1 to 11 odd #’s, 13 to 17 ALL, & 19 Prepare for Test on Chapter 1: Points, Lines, and Planes