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# Language of Geometry

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### Language of Geometry

1. 1. Geometric Structure Basic Information
2. 2. Please go back or choose a topic from above. Introduction Instruction Examples Practice
3. 3. Geometry is a way of thinking about and seeing the world. Geometry is evident in nature, art and culture. What geometric objects do you see in this picture? Introduction Instruction Examples Practice
4. 4. Geometry is both ancient and modern. Geometry originated as a systematic study in the works of Euclid, through its synthesis with the work of Rene Descartes, to its present connections with computer and calculator technology. What geometric objects do you see in this picture? Introduction Instruction Examples Practice
5. 5. The basic terms and postulates of geometry will be introduced as well as the tools needed to explore geometry. What geometric term are you familiar with? Introduction Instruction Examples Practice
6. 6. Please go back or choose a topic from above. Introduction Instruction Examples Practice
7. 7. Three building blocks of geometry are points , lines and planes . They are considered building blocks because they are basic and undefined in terms of other figures. This is page 1 of 22 Page list Last Next Introduction Instruction Examples Practice
8. 8. <ul><li>A point is the most basic building block of geometry. </li></ul><ul><li>has no size </li></ul><ul><li>indicates location </li></ul><ul><li>represented by a dot </li></ul><ul><li>named with a capital letter. </li></ul>This is page 2 of 22 Page list Last Next Introduction Instruction Examples Practice
9. 9. <ul><li>A line is a straight, continuous arrangement of infinitely many points . </li></ul><ul><li>has infinite length but no thickness. </li></ul><ul><li>extends forever in two directions. </li></ul>This is page 3 of 22 Page list Last Next Introduction Instruction Examples Practice
10. 10. A line is named with two identified points on the line with a line symbol (double-headed arrows) placed over the letters; or by a single, lower case script letter. This is page 3 of 22 Page list Last Next Introduction Instruction Examples Practice
11. 11. <ul><li>A plane has length and width but no thickness. </li></ul><ul><li>is like a flat surface the extends infinitely along its length and width. </li></ul><ul><li>represented by a four-sided figure drawn in perspective. </li></ul>This is page 4 of 22 Page list Last Next Introduction Instruction Examples Practice
12. 12. A plane is named with a script capital letter, Q. It may also be named using three points (not on the same line) that lie in the plane, such as G, F and E. This is page 5 of 22 Page list Last Next Introduction Instruction Examples Practice
13. 13. <ul><li>An axiomatic system is a way of organizing facts. </li></ul><ul><li>postulates are accepted without proof </li></ul><ul><li>theorems are truths that can be derived from postulates </li></ul>This is page 15 of 22 Page list Last Next Introduction Instruction Examples Practice
14. 14. Mathematicians accept undefined terms and definitions so that a consistent system may be built. The theorems of an axiomatic system rest on postulates and other theorems. This is page 16 of 22 Page list Last Next Introduction Instruction Examples Practice
15. 15. As with all axiomatic systems, geometry is connected with logic. This logic is typically expressed with convincing argument or proof . This is page 17 of 22 Page list Last Next Introduction Instruction Examples Practice
16. 16. <ul><li>Consider the model. Look at points A and E. </li></ul><ul><li>How many lines pass through these two points? </li></ul><ul><li>Complete the postulate: </li></ul><ul><li>Through any two points there is exactly one ______________. </li></ul>This is page 20 of 22 Page list Last Next line Introduction Instruction Examples Practice
17. 17. <ul><li>Consider the model. Look at points A, E and H. </li></ul><ul><li>How many planes pass through these three noncollinear points? </li></ul><ul><li>Complete the postulate: </li></ul><ul><li>Through any three noncollinear points there is exactly one _______________. </li></ul>This is page 21 of 22 Page list Last Next plane Introduction Instruction Examples Practice
18. 18. Collinear points are points that lie on the same line. In the figure at the right, A, B and C are collinear. A, B and D are noncollinear. Any two points are collinear . This is page 6 of 22 Page list Last Next Introduction Instruction Examples Practice
19. 19. Coplanar points are points that lie in the same plane. In the figure at the right, E, F, G, and H are coplanar. E, F, G, and J are noncoplanar. Any three points are coplanar. This is page 7 of 22 Page list Last Next Introduction Instruction Examples Practice
20. 20. When geometric figures have one or more points in common, they are said to intersect . The set of points that they have in common is called their intersection. This is page 14 of 22 Page list Last Next Introduction Instruction Examples Practice
21. 21. <ul><li>Examine the geometric model at the right. </li></ul><ul><li>Specifically, identify the places where lines intersect each other. </li></ul><ul><li>Complete the theorem: </li></ul><ul><li>The intersection of two lines is a ___________. </li></ul>This is page 18 of 22 Page list Last Next point Introduction Instruction Examples Practice
22. 22. <ul><li>Consider the model. </li></ul><ul><li>Specifically, identify the places in the diagram where planes intersect each other. </li></ul><ul><li>Complete the postulate: </li></ul><ul><li>The intersection of two planes is a _______________. </li></ul>This is page 19 of 22 Page list Last Next line Introduction Instruction Examples Practice
23. 23. Please go back or choose a topic from above. Introduction Instruction Examples Practice
24. 24. <ul><li>Through any two points there is exactly one _______. </li></ul><ul><li>Through any three noncollinear points there is exactly one ______. </li></ul><ul><li>The intersection of two lines is a _______. </li></ul><ul><li>The intersection of two planes is a ______. </li></ul>Our First 3 Postulates and a Theorem POINT LINE LINE PLANE
25. 25. Please go back or choose a topic from above. Introduction Instruction Examples Practice
26. 26. Example 1 Back to main example page <ul><li>Name the intersection of plane ABDC and plane YZDB. </li></ul><ul><li>How many lines drawn in the figure contain point Z? Enumerate. </li></ul><ul><li>How many planes drawn in the figure contain line BY? Enumerate. </li></ul><ul><li>True or false: Two planes intersect in exactly one point. Explain. </li></ul>
27. 27. Example 2 <ul><li>Classify each statement as true or false . Explain each. </li></ul><ul><li>Two lines intersect in a plane. </li></ul><ul><li>Any three points are contained in exactly one line. </li></ul>Back to main example page
28. 28. Example 1 Back to main example page <ul><li>Name the intersection of plane ABDC and plane YZDB. </li></ul><ul><li>How many lines drawn in the figure contain point Z? </li></ul><ul><li>How many planes drawn in the figure contain line BY? </li></ul><ul><li>True or false: Two planes intersect in exactly one point. </li></ul>line BD Two – line DZ and YZ plane BAXY and plane YZDB False - line 2
29. 29. Example 2 <ul><li>Classify each statement as true or false . </li></ul><ul><li>Two lines intersect in a plane. </li></ul><ul><li>Any three points are contained in exactly one line. </li></ul>Back to main example page False - point False – only collinear points
30. 30. Please go back or choose a topic from above. Introduction Instruction Examples Practice
31. 31. With the foundational terms (point, line and plane) described, other geometric figures may be defined. This is page 8 of 22 Page list Last Next Introduction Instruction Examples Practice “ Let no one ignorant of geometry enter my door.” - Plato
32. 32. <ul><li>A segment is a part of a line that begins at one point and ends at another. </li></ul><ul><li>has two endpoints </li></ul><ul><li>named by its endpoints </li></ul><ul><li>a bar (no arrows) is drawn over the two capitalized letters </li></ul>This is page 9 of 22 Page list Last Next Introduction Instruction Examples Practice
33. 33. <ul><li>A ray is a part of a line that starts at a point and extends infinitely in one direction. </li></ul><ul><li>has one endpoint </li></ul><ul><li>named with its endpoint first </li></ul><ul><li>a single arrow is drawn over the two capitalized letters. </li></ul>This is page 10 of 22 Page list Last Next Introduction Instruction Examples Practice
34. 34. Opposite rays are two collinear rays that share a common endpoint. and are opposite rays. This is page 10 of 22 Page list Last Next Introduction Instruction Examples Practice
35. 35. The length or measure of a segment is the distance between its endpoints. e.g. the length of is PQ This is page 9 of 22 Page list Last Next Introduction Instruction Examples Practice
36. 36. Segment with equal length are said to be congruent (  ). If AB = CD, then . This is page 9 of 22 Page list Last Next Introduction Instruction Examples Practice
37. 37. B is between A and C iff they are collinear and AB + BC = AC. The midpoint of a segment is the point that divides the segment into two congruent segments. In the figure, DE = EF. This is page 9 of 22 Page list Last Next Introduction Instruction Examples Practice
38. 38. A segment bisector is a segment, ray, line or plane that intersects a segment at its midpoint. A perpendicular bisector intersects the segment at the midpoint and is perpendicular to it. This is page 9 of 22 Page list Last Next Introduction Instruction Examples Practice
39. 39. Please go back or choose a topic from above. Introduction Instruction Examples Practice
40. 40. <ul><li>An angle is a figure formed by two rays with a common endpoint. </li></ul><ul><li>The common endpoint is the vertex of the angle . </li></ul><ul><li>The rays are the sides of the angle . </li></ul><ul><li>Angles are formed when lines, rays, or line segments intersect. </li></ul>This is page 11 of 22 Page list Last Next Introduction Instruction Examples Practice
41. 41. <ul><li>An angle divides the plane into two regions </li></ul><ul><ul><li>Interior </li></ul></ul><ul><ul><li>Exterior </li></ul></ul><ul><li>If two points, one from each side of the angle, are connected with a segment, the segment passes through the interior of the angle. </li></ul>This is page 12 of 22 Page list Last Next Introduction Instruction Examples Practice
42. 42. <ul><li>An angle is named using three points. </li></ul><ul><li>The vertex must be the middle point of the name. </li></ul><ul><li>Write  SRT or  TRS. </li></ul><ul><li>Say “angle S R T” or “angle T R S.” </li></ul><ul><li>If there is no possibility of confusion, the angle may be named  S or  1. </li></ul>This is page 13 of 22 Page list Last Next Introduction Instruction Examples Practice