Basic Euclidean Geometry

1. Basic Euclidean Geometry As we begin our study of geometry, keep in mind that these basic ideas haven’t changed in over 3000 years! All of what we know about “Earth Measure” can be explained by these ideas: What is a location / address? What is a collection of locations? What is a collection of collections?

2. Using Undefined terms and definition • A point has no dimension. It is usually represented by a small dot. A Point A Always name a point with a single capital letter…

3. Using Undefined terms and definition • A line extends in one dimension. It is usually represented by a straight line with two arrowheads to indicate that the line extends without end in two directions. In this book, lines are always straight lines. l A B Line l or AB You can name a line using any two letters on the line or a small scripted letter

4. A M C B Using Undefined terms and definition A plane extends in two dimensions. 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. Plane M or plane ABC You can name a plane using 3 non-collinear points or a capital scripted letter

5. A few basic concepts . . . • Must be commonly understood without being defined. One such concept is the idea that a point lies on a line or a plane. • Collinear points are points that lie on the same line. • Coplanar points are points that lie on the same plane.

6. Ex. 1: Naming Collinear and Coplanar Points • Name three points that are collinear Solution: D, E and F lie on the same line, so they are collinear. H G E F D

7. Ex. 1: Naming Collinear and 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. H G E F D

8. Ex. 1: Naming Collinear and Coplanar Points • Name three points that are not collinear. Solution: There are many correct answers. For instance, points H, E, and G do not lie on the same line. H G E F D

9. 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. • Consider the line AB (symbolized by AB). l Line l or AB

10. More . . . A B Segment AB • 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 B A Line l or AB

11. More . . . A B Ray AB • 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 B A Line l or AB

12. More . . . A B Ray BA • 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 B A Line l or AB

13. More . . . • 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. l B C A Line l or AB

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

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

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

17. Ex. 3: 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. M Q X P N

18. Ex. 3: Drawing 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. M Q X P N

19. a. Give two other names for PQand for plane R. b.Name three points that are collinear. Name four points that are coplanar. • Other names for PQare QPand line n. Other names for plane Rare plane SVT and plane PTV. EXAMPLE 1 Name points, lines, and planes SOLUTION

20. EXAMPLE 1 Name points, lines, and planes b. Points S, P, and Tlie on the same line, so they are collinear. Points S, P, T,and Vlie in the same plane, so they are coplanar.

21. 1. Use the diagram in Example 1. Give two other names for ST. Name a point that is not coplanar with points Q, S, and T. ANSWER TS, PT; point V for Example 1 GUIDED PRACTICE

22. a. Give another name for GH . a. Another name for GHis HG . b. The rays with endpoint Jare JE, JG, JF, and JH . The pairs of opposite rays with endpoint Jare JEand JF, and JGand JH. EXAMPLE 2 Name segments, rays, and opposite rays b. Name all rays with endpoint J . Which of these rays are opposite rays? SOLUTION

23. 2. Give another name for EF ANSWER FE 3. Are HJand JH the same ray ? Are HJand HGthe same ray? Explain. ANSWER No; the rays have different endpoints.Yes; points J and G lie on the same side of H. for Example 2 GUIDED PRACTICE