1. Geometry CH 1-2 Points Lines and Planes Warm Up On a number line, graph each inequality. 1. x ≥ 3 2. 2 ≤ x ≤ 6 3. x 0 -2024 0246 01 P15 # 1-

2. Geometry CH 1-2 Points Lines and Planes In a sequence of starts and stops, an elevator travels from the first floor to the fifth floor and then to the second floor. From there, the elevator travels to the fourth floor and then to the third floor. If the floors are 3 m apart, how far has the elevator traveled? Warm Up

3. Geometry CH 1-2 Points Lines and Planes What is the area of the shaded figure? Warm Up

4. Geometry CH 1-2 Points Lines and Planes Define Geometry Identify, name, and draw points, lines, segments, rays, and planes. Apply basic facts about points, lines, and planes. Objectives

5. Geometry CH 1-2 Points Lines and Planes undefined termpoint lineplane collinearcoplanar segmentendpoint rayopposite rays postulate Vocabulary

6. Geometry CH 1-2 Points Lines and Planes Geometry (Greek) "measurement of earth or land" The study of geometry can be broken into two broad types: plane geometry and Solid geometry

7. Geometry CH 1-2 Points Lines and Planes plane geometry: which deals with only two dimensions, it deals with objects that are flat, such as triangles and lines, that can be drawn on a flat piece of paper. Solid geometry: allows width, depth and height. The world around us is obviously three-dimensional, such as cubes and spheres.

8. Geometry CH 1-2 Points Lines and Planes Clearly, our world is three dimensional. But in the fictional story Flatland by Edwin Abbott, he speculates what living in a two-dimensional world (a plane) would be like. Surprisingly for a science fiction story, it was written in 1884, and his writing style is quaintly Victorian as a result. An excerpt from Chapter 1:..Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows... Read it online at "Flatland - A romance of many dimensions" by Edwin A. Abbott, a Square,1884;Flatland - A romance of many dimensions

9. Geometry CH 1-2 Points Lines and Planes The most basic figures in geometry are undefined terms, which cannot be defined by using other figures. The undefined terms are: Point, Line, and Plane which are the building blocks of geometry.

10. Geometry CH 1-2 Points Lines and Planes Point: A exact location or place. Usually represented by a dot. It is important to understand that a point is not a thing, No size. We indicate the position of a point by placing a dot with a pencil. This dot may have a diameter of, say, 0.2mm, but a point has no size. Points are usually named by using an upper-case single letter. If a set of points all lie in a straight line, they are called 'collinear'. If a set of points all lie on the same plane, they are called 'coplanar'.

11. Geometry CH 1-2 Points Lines and Planes Point: A exact location or place. Usually represented by a dot. It is important to understand that a point is not a thing, but a place.

12. Geometry CH 1-2 Points Lines and Planes Collinear: When a set of points all lie in a straight line.

13. Geometry CH 1-2 Points Lines and Planes Points that lie on the same line are collinear. K, L, and M are collinear. K, L, and N are noncollinear. Points that lie on the same plane are coplanar. Otherwise they are noncoplanar. M K L N

14. Geometry CH 1-2 Points Lines and Planes Line: A geometrical object that is straight, infinitely long and infinitely thin. A line is one-dimensional. It has zero width. A straight line is the shortest distance between any two points on a plane.

15. Geometry CH 1-2 Points Lines and Planes Line: In the figure below, the line PQ passes through the points P and Q, and goes off in both directions forever, and is perfectly straight. A line, strictly speaking, has no ends.

16. Geometry CH 1-2 Points Lines and Planes PLANE: A flat surface that is infinitely large and with no thickness and extends forever. It is difficult to draw planes, since the edges have to be drawn. When you see a picture that represents a plane, always remember that it actually has no edges, and it is infinitely large. pointlinePlaneSolid Zero dimensionsOne dimensionTwo dimensionsThree dimensions

17. Geometry CH 1-2 Points Lines and Planes Plane:

18. Geometry CH 1-2 Points Lines and Planes Plane:

19. Geometry CH 1-2 Points Lines and Planes Plane explained: Imagine a flat sheet of metal. Now make it infinitely large in both directions. This means that no matter how far you go, you never reach its edges. Now imagine that it is so thin that it actually has no thickness at all. This is the 'plane' in geometry. The plane has two dimensions: length and width. But since the plane is infinitely large, the length and width cannot be measured. Just as a line is defined by two points, a plane is defined by three points. Given three points that are not collinear, there is just one plane that contains all three.

20. Geometry CH 1-2 Points Lines and Planes

21. Geometry CH 1-2 Points Lines and Planes

22. Geometry CH 1-2 Points Lines and Planes Example 1: Naming Points, Lines, and Planes A. Name four coplanar points. B. Name three lines. A, B, C, D Possible answer: AE, BE, CE TEACH! Example 1

23. Geometry CH 1-2 Points Lines and Planes Line Segment: A straight line which links two points without extending beyond them. A line segment is one-dimensional. It has a measurable length, but has zero width.. The word 'segment' typically means 'a piece' of something, and here it means the piece of a full line, which would normally extend to infinity in both directions.

24. Geometry CH 1-2 Points Lines and Planes Line Segment: See the figure below. The line segment PQ links the points P and Q. The points P and Q are called the 'endpoints' of the segment.

25. Geometry CH 1-2 Points Lines and Planes RAY: A portion of a line which starts at a point and goes off in a particular direction to infinity. One way to think of a ray is a line with one end. A ray starts at a given point and goes off in a certain direction forever, to infinity. The point where the ray starts is called (confusingly) the endpoint. A ray has no measurable length, because it goes on forever in one direction.

26. Geometry CH 1-2 Points Lines and Planes A Ray On its way to infinity it may pass through one or more other points. In the figure above, the ray starts at A and also passes through B..

27. Geometry CH 1-2 Points Lines and Planes Opposite Rays: Opposite rays are two rays that both start from a common point and go off in exactly opposite directions and form a straight line. You may name a ray using the Endpoint and any other point which the ray passes through. Such as Do not forget the ray crown

28. Geometry CH 1-2 Points Lines and Planes Opposite Rays: When the two rays are opposite, the points A,Q and B are collinear, and QA and QB form a single straight line through the common endpoint Q...

29. Geometry CH 1-2 Points Lines and Planes

30. Geometry CH 1-2 Points Lines and Planes Example 2: Drawing Segments and Rays Draw and label each of the following. A. a segment with endpoints M and N. B. opposite rays with a common endpoint T. M N T

31. Geometry CH 1-2 Points Lines and Planes Draw and label a ray with endpoint M that contains N. TEACH! Example 2 MN

32. Geometry CH 1-2 Points Lines and Planes A postulate, is a statement that is accepted as true without proof. Postulates about points, lines, and planes help describe geometric properties.

33. Geometry CH 1-2 Points Lines and Planes

34. Geometry CH 1-2 Points Lines and Planes

35. Geometry CH 1-2 Points Lines and Planes Name a line that passes through two points. Example 3: Identifying Points and Lines in a Plane XY

36. Geometry CH 1-2 Points Lines and Planes Name a plane that contains three noncollinear points. TEACH! Example 3 Possible answer: plane GHF

37. Geometry CH 1-2 Points Lines and Planes Use a dashed line to show the hidden parts of any figure that you are drawing. A dashed line will indicate the part of the figure that is not seen. HINT

38. Geometry CH 1-2 Points Lines and Planes Example 4: Representing Intersections A. Sketch two lines intersecting in exactly one point. B. Sketch a figure that shows a line that lies in a plane.

39. Geometry CH 1-2 Points Lines and Planes TEACH! Example 4 Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane.

40. Geometry CH 1-2 Points Lines and Planes Lesson Quiz: Part I 1. Two opposite rays. 3. The intersection of plane N and plane T. 4. A plane containing E, D, and B. 2. A point on BC.

41. Geometry CH 1-2 Points Lines and Planes Lesson Quiz: Part II 5. a line intersecting a plane at one point 6. a ray with endpoint P that passes through Q Draw each of the following.

42. Geometry CH 1-2 Points Lines and Planes Lesson Quiz: Part I 1. Two opposite rays. 3. The intersection of plane N and plane T. 4. A plane containing E, D, and B. 2. A point on BC. CB and CD Possible answer: D Possible answer: BD Plane T

43. Geometry CH 1-2 Points Lines and Planes Lesson Quiz: Part II 5. a line intersecting a plane at one point 6. a ray with endpoint P that passes through Q Draw each of the following.