Chapter 1: Points, Lines, Planes, and Angles


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1 Chapter 1: Points, Lines, Planes, and Angles (page 1) 11: 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 from A and B because they are equally distant from A and B. 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
2 12: 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
3 SPACE: the set of points. GEOMETRIC FIGURE: a of points. [figure 1] [figure 2] V A B 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 lie in the same plane. example from figure 2:
4 INTERSECTION (of two or more figures): the set of to all figures. that are common examples: (1) Lines (2) Planes Assignment: Written Exercises, pages 7 to 9: 1 to 35 odd # s Prepare for Quiz on Lessons 11 & 12
5 13: Segments, Rays, and Distance (page 11) Point B is A and C if it lies on line AC ( ). A B C SEGMENT: consists of points A and C and all points that are A and C. example: A C RAY: consists of AC and all other points P such that C is A and P. example: A C P OPPOSITE RAYS: two rays that have the same and form a line. example: X O Y Geometry and Algebra are brought together with the line. A B C D E F G H I J Every point is paired with a. Every number is paired with a. 1  to  1 Correspondence A corresponds to B corresponds to or A!, or B!, etc. LENGTH (of a segment): the between its endpoints. AB means the of AB or the between points A & B. AB = a  b = b  a examples: CH = AJ = BF =
6 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 = x1, 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 B expresses a relationship between numbers. D! AB CD 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
7 14: Angles (page 17) ANGLE: a figure formed by 2 rays that have the same. 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 degrees. Right Angle: measure equals degrees. Obtuse Angle: measure is between and degrees. Straight Angle: measure equals 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
8 How many angles are shown in the diagram below? Name the different angles: X 2 O 3 Y Z POSTULATE 3 PROTRACTOR POSTULATE 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 = x! y. Q P A O B POSTULATE 4 ANGLE ADDITION POSTULATE 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.
9 CONGRUENT ANGLES: angles that have example: measure. 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) Yes or No Yes or No Yes or No (4) (5) (6) Yes or No Yes or No Yes or No BISECTOR of an ANGLE: the ray that divides the angle into 2 congruent angles. example: O X Z m!xoy m!yoz then!xoy!yoz! OY "XOZ
10 MORE EXAMPLES: In diagram, OB bisects!aoc. A 2 B 3 1 D O C (1) m! 1 = 2x +5 and m! 2 = 3x 12. m! 1 = m! 2 = m! 3 = (2) 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 13 & 14
11 15: Postulates and Theorems Relating Points, Lines, & Planes (page 22) The phrases, exactly one and one and only one imples and. POSTULATE 5 A line contains at least points; A plane contains at least Space contains at least points not all in one line; points not all in one plane. POSTULATE 6 Through any two points there is exactly line. POSTULATE 7 Through any three points there is at least 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 11 If two lines intersect, then they intersect in exactly point. NO PROOF  example: THEOREM 12 Through a line and a point not in the line there is exactly plane. NO PROOF  example: THEOREM 13 If two lines intersect, then exactly plane contains the lines. NO PROOF  example:
12 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:
13 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
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