1. Geometry Ms. Bateman 2010 - 2011 1.1 Points, Lines and Planes

2. Objectives Understand, identify, and model the basic undefined terms of geometry – point, line, and plane. Understand, identify, and model the basic undefined terms of geometry – point, line, and plane. Understand, identify, and model the basic definitions of geometry – collinear, coplanar Understand, identify, and model the basic definitions of geometry – collinear, coplanar Identify the intersections of lines and planes in space. Identify the intersections of lines and planes in space.

3. Undefined Terms In geometry, some words such as point, line and plane are undefined terms. They are only explained by using examples and descriptions. They are not formally defined but they can be used to define other words.

4. Undefined Term - Point A point is a location without shape or size. It has no dimensions. It is named by a capital letter and represented by a small dot. Points in space are used to form windows through which a rocket must pass in order to stay on the right path. A Point A

5. Undefined Term - Line A line extends in one direction - length. It contains points and has no width or thickness. It is represented by a straight line with two arrowheads to indicate that the line extends without end in two directions. A B l Line l or AB

6. Undefined Term - Plane A plane extends in two dimensions – length and width. It is usually represented by a shape that looks like a tabletop or wall. You must imagine that the plane extends without end even though the drawing of a plane appears to have edges. A B C M Plane M or plane ABC

7. Defined Terms... A definition uses known words to describe a new word. Examples: Collinear points, coplanar points, line segment, ray, opposite rays, parallel lines, and perpendicular lines

8. Collinear & Coplanar Points Collinear points are points that lie on the same line. Collinear points are points that lie on the same line. There is exactly one line through any two points. There is exactly one line through any two points. Coplanar points are points that lie on the same plane. Coplanar points are points that lie on the same plane. The intersection of two planes is a line. The intersection of two planes is a line.

9. Naming Collinear Points Name three points that are collinear Solution: D, E and F lie on the same line, so they are collinear. G D E F H

10. Naming Coplanar Points Name four points that are coplanar. Solution: D, E, F, and G lie on the same plane, so they are coplanar. Also D, E, F, and H are coplanar; although, the plane containing them is not drawn. G D E F H

11. Naming Collinear & Coplanar Points Name three points that are not collinear. Solution: There are many correct answers. A, B, and D B, C, and D A, B, and E B, C, and E A, C, and D A, C, and E D A B C E

12. More... Another undefined concept in geometry is the idea that a point on a line is between two other points on the line. You can use this idea to define other important terms in geometry. Another undefined concept in geometry is the idea that a point on a line is between two other points on the line. You can use this idea to define other important terms in geometry. Consider the line AB (symbolized by AB). Consider the line AB (symbolized by AB). l Line l or AB

13. Definition... The line segment or segment AB (symbolized by AB) consists of the endpoints A and B, and all points on AB that are between A and B. The line segment or segment AB (symbolized by AB) consists of the endpoints A and B, and all points on AB that are between A and B. l Line l or AB A A B B Segment AB

14. Definition... The ray AB (symbolized by AB) consists of the initial point A and all points on AB that lie on the same side of A as point B. The ray AB (symbolized by AB) consists of the initial point A and all points on AB that lie on the same side of A as point B. l Line l or AB A A B B Ray AB

15. More... Note that AB is the same as BA and AB is the same as BA. However, AB and BA are not the same. They have different initial points and extend in different directions. Note that AB is the same as BA and AB is the same as BA. However, AB and BA are not the same. They have different initial points and extend in different directions. l Line l or AB A A B B Ray BA

16. More... If C is between A and B, then CA and CB are opposite rays. If C is between A and B, then CA and CB are opposite rays. Like points, segments and rays are collinear if they lie on the same line. So, any two opposite rays are collinear. Segments, rays and lines are coplanar if they lie on the same plane. Like points, segments and rays are collinear if they lie on the same line. So, any two opposite rays are collinear. Segments, rays and lines are coplanar if they lie on the same plane. l Line l or AB A C B

17. Drawing lines, segments and rays Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. J K L Draw J, K and L Then draw JK

18. Ex. 2: Drawing lines, segments and rays Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. J K L Draw KL

19. Drawing lines, segments and rays Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. J K L Draw LJ

20. Drawing Opposite Rays Draw two lines. Label points on the lines and name two pairs of opposite rays. Solution: Points M, N, and X are collinear and X is between M and N. So XM and XN are opposite rays. P M Q N X

21. Drawing Opposite Rays Draw two lines. Label points on the lines and name two pairs of opposite rays. Draw two lines. Label points on the lines and name two pairs of opposite rays. Solution: Points P, Q, and X are collinear and X is between P and Q. So XP and XQ are opposite rays. P M Q N X

22. Sketching Intersections of Lines and Planes Two or more lines intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common. Two or more lines intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common. Modeling Intersecting Planes Modeling Intersecting Planes Complete Geometry Activity on page 8 - Use two index cards and label them as shown.

23. Sketching intersections Sketch the figure described. Sketch the figure described. a. A line that intersects a plane in one point  Draw a plane and a line.  Emphasize the point where they meet.  Dashes indicate where the line is hidden by the plane

24. Sketching intersections Sketch the figure described. Sketch the figure described. b. Two planes that intersect in a line  Draw two planes.  Emphasize the line where they meet.  Dashes indicate where one plane is hidden by the other plane.