2. 1.1 Intro to Geometry

3. Geometry The word " geometry " comes from two Greek words geo and metron meaning "earth measuring." Geometry was extremely important to ancient societies and was used for surveying, astronomy, navigation, and building. Geometry, as we know it is actually known as Euclidean geometry which was written well over 2000 years ago in Ancient Greece by Euclid, Pythagoras, Thales, Plato and Aristotle just to mention a few. The most fascinating and accurate geometry text was written by Euclid, and was called Elements. Euclid's text has been used for over 2000 years!

4. Millau Bridge Sir Norman Foster Point, Lines, Planes, Angles Fallingwaters Frank Lloyd Wright Millenium Park Frank Lloyd Wright Components of Geometry Part 1

5. GEOMETRY’S UNDEFINED TERMS POINTLINEPLANE

6. POINT A point is a_______, or a_____.

7. POINT A point is not a physical object – you can not touch it, feel it, or even see it.

8. POINT A point has no size.

9. POINT A point can be represented by a dot. The size that you make the dot is not important because a point has no size. Make it just large enough to see.

10. POINT A point is named by using an “upper-case” “block” letter. EXAMPLE: P This figure would be called “point P”.

11. POINT EXAMPLE 2: X This figure would be called “point X”.

12. LINE A line is a set of points that extends infinitely in 2 opposite directions. What does “infinitely” mean ? NEVER-ENDING (Goes on forever.)

13. LINE EXAMPLE : Why does it have arrows on both ends ? TO SHOW THAT IT EXTENDS INFINITELY IN BOTH (OPPOSITE) DIRECTIONS

14. NAMING A LINE There are 2 ways: 1. Use the names of (any) two points on the line. Example: P X What name(s) can we give to this line? Line ___or line____

15. NAMING A LINE There are 2 ways: 2. Use a “lower-case” “cursive” letter (located near one of the arrows). Example: What name can we give to this line?

16. NAMING A LINE There are 2 ways: 2. Use a “lower-case” “cursive” letter (located near one of the arrows). 1. Use the names of (any) two points on the line. EXAMPLE : line PX (OR USE A “SYMBOL” FOR LINE AND WRITE IT ABOVE THE LETTERS.) PX EXAMPLE: line ____

17. How many names does this line have? R G K There are 7 different names that you could use to name this line. They are:

18. Let’s try again. How many names does this line have? F N X t There are 7 different names that you could use to name this line. They are:

19. ONE MORE TIME. How many names does this line have? T Y N There are 12 different names that you could use to name this line. They are: C

20. PLANE A plane is a “flat surface” – a set of points that extends infinitely in 2 dimensions. What does “dimension” mean ? ________,_______,__________. A plane is infinitely long and infinitely wide, but it has no height (or depth).

21. PLANE A plane can be drawn (represented) by a four sided figure known as a parallelogram.

22. PLANE A plane can be named by a single “upper-case” “block” letter (written near a corner of the plane). Q What is the name of this plane? Plane Q or □Q

23. What is the name of this plane? R N T F ANSWER: _______ WHY CAN’T IT BE CALLED PLANE R or T or N ? http://search.mywebsearch.com/mywebse arch/video.jhtml?searchfor=lines+planes+ point+geometry&n=77fce313&ss=sub&st= kwd&ptb=E3835453-B3EB-46F1-B69F- 564B8FD71B8B&tpr=sbt http://search.mywebsearch.com/mywebse arch/video.jhtml?searchfor=lines+planes+ point+geometry&n=77fce313&ss=sub&st= kwd&ptb=E3835453-B3EB-46F1-B69F- 564B8FD71B8B&tpr=sbt

24.  Descriptions of Defined Terms  Space  General Terms  Congruent  Similar  Equal  Union  Intersection

25.  Space Refers to the set of all points. Space goes on forever in every direction, and therefore has length, width, and depth. Space has no special notations. It is simply referred to as space. Space contains at least 4 points that are not all on the same plane.

26.  General Terms  Congruent  Similar  Equal  Union  Intersection

27. CCongruent  or congruent shapes are the same shape and size; therefore, after some movement of the shapes they can be made to fit exactly on top of one another.

28. CCongruent  or congruent shapes are the same shape and size; therefore, after some movement of the shapes they can be made to fit exactly on top of one another.

29.  Similar  or similar shapes are the same shape, but can be different sizes; thus congruent shapes are also similar shapes, but similar shapes are not necessarily congruent shapes.

30.  Equal = or equal can apply to sets of points being exactly the same set or to numerical measurements being exactly the same number values.

31.  Union  or union refers to putting all of the points together and describing the result.

32.  Intersection  or intersection refers to describing only those points that are common to all sets involved in the intersection or to describing the points where indicated shapes touch.

33. Symbolized by 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. AB A is called the initial point Ray AB extends in one direction without end. Let’s look at a ray:

34. Not all symbols are created equal! Not all symbols are created equal! Symbol alert! is the same as BUT… AB AB

35. is not the same as Symbol alert!! AB AB Initial point 1 st 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.

36. If C is between A and B, If C is between A and B, Opposite Rays AB C then and are opposite rays. C is the common initial point for the rays!

37. Rays are important because they help us define something very important in 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. 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. Angles 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:

38. 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°. 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

40. Two or more geometric figures intersect if they have one or more points in common. 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 The intersection of the figures is the set of points the figure has in common Intersections of lines and planes How do 2 line intersect? How do 2 planes intersect? What about a line and a plane?

41. Modeling Intersections To think about the questions on the last slide lets look at the following… To think about the questions on the last slide lets look at the following… B A E Two 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