Basics of geometry
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Points, lines, and planes are the undefined terms in geometry that form its foundations. A point has no dimensions and marks a location in space, a line extends in one dimension, and a plane extends in two dimensions. The basic elements of 2D space are points, lines, line segments, rays, angles, and their intersections. Rays and angles are defined using points and lines, with rays having a starting point and angles consisting of two rays with the same starting point.
Basics of geometry
- 1. Basics of Geometry Basics of Geometry POINTS! LINES! PLANES!
- 2. Basics of Geometry • What are the undefined terms in geometry? • What concepts present the foundations of geometry? • What are the basic elements of 2D space?
- 3. Basics of Geometry The terms points, lines, and planes are the foundations of geometry, but… point, line, and plane are all what we call undefined terms. How can that be? Well, any definition we could give them would depend on the definition of some other mathematical idea that these three terms help define. In other words, the definition would be circular! Undefined Terms?
- 4. Basics of Geometry • Has no dimension • Marks a location in space • Usually represented by a small dot Point A The above is called point A. Note the point is represented with a capital letter.
- 5. Basics of Geometry • Extend in one dimension. • Represented with straight line with two arrowheads to indicate that the line extends without end in two directions. Line l B A This is Line l, (using the lower case script letter) or symbolically we call it NOTICE: The arrowheads are in both directions on the symbol
- 6. Basics of Geometry Plane • Extend in two dimensions. • Represented by a slanted 4 sided figure, but you must envision it extends without end, even though the representation has edges. A B C M This is Plane M or plane ABC (be sure to only use three noncollinear points when naming a plane)
- 7. Basics of Geometry Collinear • Collinear points are points that lie on the same line. l C A B Points A, B and C are collinear.
- 8. Basics of Geometry Coplanar • Coplanar points are points that lie on the same plane. A B C Points A, B and C are coplanar.
- 9. Basics of Geometry Line Segment Let’s look at the idea of a point in between two other points on a line. Here is line AB, or recall symbolically A B The line segment does not extend without end. It has endpoints, in this case A and B. The segment contains all the points on the line between A and B A B Notice the difference in the symbolic notation! This is segment
- 10. Basics of Geometry Symbolized by Ray The initial point is always the first letter in naming a ray. Notice the difference in symbols from both a line and segment. A B A is called the initial point Ray AB extends in one direction without end. Let’s look at a ray:
- 11. Basics of Geometry Not all symbols are created equal! Symbol alert! is the same as is the same as BUT… A B A B
- 12. Basics of Geometry is not the same as Symbol alert!! A B A B Initial point 1st The ray is different! Notice that the initial point is listed first in the symbol. Also note that the symbolic ray always has the arrowhead on the right regardless of the direction of the ray.
- 13. Basics of Geometry If C is between A and B, Opposite Rays A BC then and are opposite rays. C is the common initial point for the rays!
- 14. Basics of Geometry Angles Rays are important because they help us define something very important in geometry…Angles! An angle consists of two different rays that have the same initial point. The rays are sides of the angles. The initial point is called the vertex. vertex sides A B C Notation: We denote an angle with three points and symbol. The middle point is always the vertex. We can also name the angle with just the vertex point. This angle can be denoted as:
- 15. Basics of Geometry Straight angle m A = 180° Obtuse angle 90°< m A < 180° Acute angle 0°< m A < 90° Classifying Angles Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0° and less or equal to 180°. A A Right angle m A = 90° A A
- 16. Basics of Geometry Intersections of lines and planes • Two or more geometric figures intersect if they have one or more points in common. • The intersection of the figures is the set of points the figure has in common Think! ! How do 2 line intersect? How do 2 planes intersect? What about a line and a plane?
- 17. Basics of Geometry Modeling Intersections To think about the questions on the last slide lets look at the following… B A ETwo lines intersect at a point, like here at point A. F Line BF is the intersection of the planes G and H. G H Point E is the intersection of plane H and line EC D C
- 18. Basics of Geometry Something to think about… You have just finished the first section in Geometry! This is a very important section because it lays the foundation for the rest of the year! Much of the vocabulary you will encounter in this course will have its foundation in the ideas presented in this lesson. Can you name the three undefined terms in geometry? Do you know the difference between and obtuse and straight angle? Can you sketch the intersection of a plane and a line? How about two planes? Can you visualize the intersection of two planes? How about three? The classfun and homefun provided will help you in developing a better understanding of the concepts!
- 19. Basics of Geometry Draw 3 noncollinear points J, K, L, then draw: Quick Quiz!!! L K J When ready click to see answers!
- 20. Basics of Geometry Quick Quiz !!!! 1. Name 3 points that are collinear. Points D, F and B lie on the same line, therefore they are collinear. 2. Name 4 points that are coplanar. If you answered D, F, B, and G are coplanar, then you are correct. It is harder to see because the plane is not drawn, but points D, F, B and M are also coplanar. 3. Name 3 points that are not collinear. There are many correct answers, D, F, M are not coplanar. M G D F B
- 21. Basics of Geometry Use the diagram below to answer the following questions. a. Name the type of angle. Acute b. Name the vertex. R c. Name the sides of the angle. d. Name the angle three different ways. Quick Quiz !!!! R S T