1. 1. (Collinearity and Coplanarity) Properties of Geometric Figures Mai Nicole R. Olaguer III-B BSMT
2. 2. What are the undefined terms in Geometry?
3. 3. POINT LINE PLANE
4. 4. Point A point represents position only; it has a zero size (zero length, zero width, and zero height) How to name: Capital letter Read as: Point A A
5. 5. Point A b P H 1 E G f
6. 6. Line A line (straight line) is a connected set of infinite points. It extends infinitely far in two opposite directions. It has infinite length, zero width, and zero height.
7. 7. Line How to name: Two capital letters with a symbol ( ) above them Lower case letter AB (line AB), BA (line BA), or (line ) A B l l l
8. 8. Line P R l m Q a B 1
9. 9. Plane A plane is an infinite set of points forming a connected flat surface extending infinitely far in all directions. It has infinite length, infinite width, and zero height.
10. 10. Plane How to name: Single capital letter Read as: Plane M M
11. 11. Plane l Q 4
12. 12. 1. Wire 2. Yarn 3. Table Top 4. Salt 5. Money 6. Floor 7. Tip of the ballpen 8. Edge of the ruler 9. Tip of the needle 10. Sheet of paper Identify what undefined term is the following:
13. 13. Definition cA B D Collinear points are points that lie on the same line. If there is no line on which all the points lie, then they are non-collinear points.
14. 14. Definition Coplanar points are three or more points that lie on the same plane. The points which do not lie in the same plane are non-coplanar points. A B D
15. 15. The Line Postulate Postulate 1 BA For any two points, there is exactly one line that contains both points.
16. 16. The number of Points Postulate Postulate 2 A plain contains at least three non-collinear points. A B
17. 17. The number of Points Postulate Postulate 2 A space contains at least four non- collinear points. A B D
18. 18. Any three points lie in at least one plane and any three non- collinear points lie in exactly one plane. A B The Plane Postulate Postulate 3
19. 19. The Plane Intersection Postulate Postulate 4 If two planes intersect, then their intersection is a line. W X Y Z A
20. 20. The Flat Plane Postulate Postulate 5 If two points of a line lies on the plane, the entire line lies on the plane. A B
21. 21. The Line Intersection Theorem Theorem 1 If two lines intersect, then their intersection is exactly one point.A B
22. 22. The Line-Plane Intersection Theorem Theorem 2 Given a plane and a line not on the plane, their intersection is one and only one point.A B M
23. 23. The Line-Point Theorem Theorem 2 Given a line and a point not on the line, there is exactly one plane that contains them.A B M
24. 24. The Lines-Plane Theorem Theorem 2 Given two intersecting lines, there is exactly one plane that contains the two lines.A B M
25. 25. 1.Can two lines intersect in two points? No, If they did, then there would be two different lines containing the two points. This contradicts the Line Postulate. 2. What are the possible intersections of a line and a plane? Draw a picture of each. - Exactly one point - The line itself Problems:
26. 26. 3. What is/are the possible intersections of two planes? Line 4. True or False. An angle is contained in exactly one plane. True, the angle is made up of two rays which determine two intersecting lines. By the Lines-Plane Theorem, there is exactly one plane that contains the two intersecting lines. Problems:
27. 27. 1. An infinite number of lines in a point 2. An infinite number of points in a line 3. Lines in three non-collinear points 4. Two planes intersect in one line 5. Four non-coplanar points 6. Plane and a line intersect at one point 7. An infinite number of planes in a point 8. Two coplanar lines 9. Four non-collinear points 10. Intersecting lines in a plane Draw:
28. 28. Collinear points are points that lie on the same line. Coplanar points are three or more points that lie on the same plane. Postulate 1. The line Postulate: For any two points, there is exactly one line that contains both points. Postulate 2. The number of Points Postulate: A plain contains at least three non- collinear points. Generalization:
29. 29. Postulate 3. The Plane Postulate: Any three points lie in at least one plane and any three non-collinear points lie in exactly one plane. Postulate 4. The Plane Intersection Postulate: If two planes intersect, then their intersection is a line. Postulate 5. The Flat Plane Postulate: If two points of a line lie on a plane, the entire line lies on the plane. Theorem 1. The Line Intersection Generalization:
30. 30. Theorem 2. The Line-Plane Intersection Theorem: Given a plane and a line not on the plane, their intersection is one and only one point. Theorem 3. The Line-Point Theorem: Given a line and a point not on the line, there is exactly one, plane that contains them. Theorem 4. The Lines-Plane Theorem: Given two intersecting lines, there is exactly one plane that contains the two Generalization:
31. 31. Modify: True or False. 1.Intersecting lines are always coplanar. 2.Three points are never coplanar. 3.A line and a point not on the line lie in exactly one plane. 4.Four points are sometimes coplanar. 5.Two planes can intersect in exactly one point. 6.The intersection of a line and a plane can be an empty set, a point, or a lie. 7.Two lines can intersect in exactly one point. HAPPY QUIZ:
32. 32. Study about the measurement of angles. Difference between these angles: 1. Acute Angle 2. Right Angle 3. Obtuse Angle ASSIGNMENT:
33. 33. END