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Transcript
OTCQ 091709 Are the integers a closed set under division? OTCQ 091709 Are the integers a closed set under division? No -1/2, ¾, -9/7 … Remember a + or - integer has a decimal of .0000…. Aim 1-4 and 1-5 How do we define midpoints, bisectors, rays and angles? NY GG 31, GG 32, GG 33, GG 35, GG42, GG 43 Objective1 SWBAT define midpoints, bisectors, rays and angles? Geometry • Point – An exact location in space • Line – An endless collection of points along a straight path • Line segment – Part of a line that has two endpoints . A Geometry Ray Part of a line that has one endpoint and extends endlessly in the other direction Plane An endless, flat surface that is named by any three points not on the same line. .B .C .A Lines • A line has infinite length, zero width, and zero height. Any two points on the line can be used to name it. The symbol ↔ written on top of two letters is used to denote that line. A line may also be named by one small letter l. Line Segment • A line segment does not extend forever. A line segment has two endpoints. We write the name of a line segment with endpoints A and B as AB . B A Midpoint and Bisector of a Segment • MIDPOINT-A point on the line segment that cuts the segment into two congruent pieces. • BISECTOR OF A SEGMENT-A line, segment, ray, or plane that intersects the segment at its midpoint. Ray • We may think of a ray as a "straight" line that begins at a certain point and extends forever in one direction. • The point where the ray begins is known as its endpoint. • We write the name of a ray with endpoint A and passing through a point B as . Angles An angle is formed when two rays have the same endpoint. This endpoint is called the vertex. The two rays that form the angle are called sides. Angles, Cont. • We name an angle by using a point on each ray and the vertex. The angle below may be specified as angle ABC or as angle CBA; you may also see this written as ABC or as CBA. Practice Name each figure 1. 3. 2. 4. 5. Angles • There are four types of angles – Right angle – Straight angle – Acute angle – Obtuse angle Right Angles • Forms a square corner • Forms a 90 degree angle. 90 degrees – Use a corner of your index card to make sure you a have a right angle Straight Angle • Forms a straight line • Angle is 180 degrees 180 degrees – Use the edge of your index card to make sure you have a straight angle. Acute Angles • Forms an angle that is less than a right angle • Angle is less than 90 degrees Obtuse Angles • Form an angle that is more than a right angle • Angle is more than 90 degrees Review Name each picture • Ray • Parallel lines 1. 2. • Line 3. Review Name each picture • Acute angle 4. 5. 6. • Line segment • Point Degrees: Measuring Angles • We measure the size of an angle using degrees with the Protractor Obtuse or Acute? 1. 3 2. 4. Postulates • A statement that is accepted without proof. • Usually these have been observed to be true but cannot be proven using a logic argument. Postulates Relating Points, Lines, and Planes • Postulate 5: A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane. Postulates Relating Points, Lines, and Planes • Postulate 6: Through any two points there is exactly one line. Postulates Relating Points, Lines, and Planes • Postulate 7: Through any three points there is at least one plane (if collinear), and through any three non-collinear points there is exactly one plane. Postulates Relating Points, Lines, and Planes • Postulate 8: If two points are in a plane, then the line that contains the points is in that plane. A . B . Postulates Relating Points, Lines, and Planes • Postulate 9: If two planes intersect, then their intersection is a line. Theorems • Theorems are statements that have been proven using a logic argument. • Many theorems follow directly from the postulates. Theorems Relating Points, Lines, and Planes • Theorem 1-1: If two lines intersect, then they intersect in exactly one point. • Theorem 1-2: Through a line and a point not in the line there is exactly one plane. • Theorem 1-3: If two lines intersect, the exactly one plane contains the lines. Points • A point represents position only; it has zero size (that is, zero length, zero width, and zero height). Types of Points • Points that lie on the same line are called collinear points. If there is no line on which all of the points lie, then they are non-collinear points. Collinear> Non-collinear> Types of Lines • PARALLEL LINES- two lines that are always the same distance apart, and will never intersect. Parallel can be abbreviated as ||. An example of parallel lines is on the Italian flag. Lines a and b on the flag are parallel. Types of Lines • SKEW LINES - two lines that do not intersect, and are not parallel. Skew lines are always non-coplanar. An overpass on a highway is an excellent example of skew lines. • This only occurs when you consider lines in 3 dimensional space. Plane • A plane may be considered as an infinite set of points forming a connected flat surface extending infinitely far in all directions. • A plane has infinite length, infinite width, and zero height (or thickness). It is usually represented in drawings by a four-sided figure. • A single capital letter is used to denote a plane. The word plane is written with the capital letter so as not to be confused with a point. Relationships to Planes • COPLANAR - on the same plane. Points or objects may not be collinear, but if they lie in the same plane they are coplanar. NONCOPLANAR - any number of points not lying in the same plane.