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Points, Lines, Planes 1-3
Quiz	 What is a conjecture? Give a counterexample: The product of 2 positive numbers is greater than either number. Use inductive reasoning:  		2, 6, 7, 21, 22, 66 67, …. Make a conjecture about the sum of the first 500 positive even numbers.
Collinear and Noncollinear Points Points that are on the same line are called collinear points. If a single line cannot be drawn through all the points, then the points are noncollinear. W U E F C D X V Collinear points Noncollinear points
Collinear Points Are points E, F, and C collinear? Can you label line m in three ways? Are points E, F, and D collinear? Are points F, P, and C collinear? n m C E F P D l
Planes Intersecting lines are lines that cross (intersect) at exactly one point. Parallel lines are lines that do not cross They have no points in common. Skew lines are not parallel and they do not intersect.  They lie in different planes.
Planes List three different names for the plane represented by the top of the box. G H F E C A B
Basic Postulates of Geometry Postulate 1-1:  Through any two points there is exactly one line.
Basic Postulates of Geometry Postulate 1-2 If two lines intersect, then they intersect in exactly one point.
Basic Postulates of Geometry Postulate 1-3 If two planes intersect,  then they intersect in  exactly one line. R S W T
Basic Postulates of Geometry Postulates1-4 Through any three noncollinear points there is exactly one plane.
Homework			Prentice Hall Geometry Page 19- 20  # 2- 32 even 16 problems
Angles
QUIZ	 What is the answer to number….. 6 8 14 20 28 30
Segments A segment is a part of a line with two endpoints. R B Read it as: “Segment RB” or “Segment BR” Write it as: RB or BR
Rays A ray consists of an endpoint and all the points of a line on one side of the endpoint. C D Read it as: “Ray CD” (the order does matter) Write it as: CD
Vocabulary An angle has two sides and a vertex. The sides of the angles are rays. The rays share a common endpoint (the vertex) Angles are measured in units called degrees.
Right Angles Forms a square corner Forms a 90 degree angle. 90 degrees
Straight Angle Forms a straight line Angle is 180 degrees 180 degrees
Types of Angles When lines intersect to form right angles, then they are classified as perpendicular lines.
What do we use to help us? A protractor Here is a standard protractor like you  use in the classroom.
When we use a protractor, we need to line it up correctly. You need to make sure the protractor is lined up correctly. Is this ready to measure the angle?
Were you right.................it wasn’t Look for the upside down ‘T’ in the middle of the straight line on your protractor.   This needs to be exactly on the vertex of your angle.
We need to remember..... It doesn’t  matter which  way round the  angle is, you  ALWAYS need  to line the upside down ‘T’ to the vertex of the angle.
Now you are ready. Read from the 0°, and follow the inner set of numbers.
Homework	 Page 25 # 1-15  Homework			Prentice Hall Geometry
QUIZ	 1. When two lines are skew, it means three things. What are those three things?  2.  Demonstrate how to label a line, plane, ray, and segment. 3. An angle is made up of two ____ sharing a common point called _____.
4. The term skew is a Middle English word meaning "to escape."  Skew lines cannot be contained in one plane. Therefore, they have "escaped" a plane.  What is something in your life that you have skewed from,  for the better?
Labeling Angles Must have three points. One on each ray and the vertex. B Labeled as:  ∠ BCD or ∠DCB C D
Complementary Angles If the sum of the measures of two angles is exactly 90º then the angles are complementary.
Supplementary Angles If the sum of the measures of two angles is exactly 180º then the angles are supplementary.
Vertical Angles Two angles whose sides are opposite rays.
Vertical Angles Vertical angles are congruent Congruent means same or equal. In this picture, <1 and <3 are vertical angles. What is the measure of <2? <2 is 120º because it is congruent to the vertical angle across from it.
Adjacent Angles Two coplanar angles with a common side, a common vertex, and no common interior points. ∠ABC is adjacent to ∠CBD A C B D
Review!		Name each picture.
Postulates.	 Angle Addition Postulate If point B is in the interior of ∠AOC,  then  m ∠AOB + m ∠BOC = m ∠AOC. A B O C
Postulates. Angle Addition Postulate If ∠AOC is a straight angle,  then  m ∠AOB + m ∠BOC = 180. B A O C
Homework			Prentice Hall Geometry

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Angles

  • 2. Quiz What is a conjecture? Give a counterexample: The product of 2 positive numbers is greater than either number. Use inductive reasoning: 2, 6, 7, 21, 22, 66 67, …. Make a conjecture about the sum of the first 500 positive even numbers.
  • 3. Collinear and Noncollinear Points Points that are on the same line are called collinear points. If a single line cannot be drawn through all the points, then the points are noncollinear. W U E F C D X V Collinear points Noncollinear points
  • 4. Collinear Points Are points E, F, and C collinear? Can you label line m in three ways? Are points E, F, and D collinear? Are points F, P, and C collinear? n m C E F P D l
  • 5. Planes Intersecting lines are lines that cross (intersect) at exactly one point. Parallel lines are lines that do not cross They have no points in common. Skew lines are not parallel and they do not intersect. They lie in different planes.
  • 6. Planes List three different names for the plane represented by the top of the box. G H F E C A B
  • 7. Basic Postulates of Geometry Postulate 1-1: Through any two points there is exactly one line.
  • 8. Basic Postulates of Geometry Postulate 1-2 If two lines intersect, then they intersect in exactly one point.
  • 9. Basic Postulates of Geometry Postulate 1-3 If two planes intersect, then they intersect in exactly one line. R S W T
  • 10. Basic Postulates of Geometry Postulates1-4 Through any three noncollinear points there is exactly one plane.
  • 11. Homework Prentice Hall Geometry Page 19- 20 # 2- 32 even 16 problems
  • 13. QUIZ What is the answer to number….. 6 8 14 20 28 30
  • 14. Segments A segment is a part of a line with two endpoints. R B Read it as: “Segment RB” or “Segment BR” Write it as: RB or BR
  • 15. Rays A ray consists of an endpoint and all the points of a line on one side of the endpoint. C D Read it as: “Ray CD” (the order does matter) Write it as: CD
  • 16. Vocabulary An angle has two sides and a vertex. The sides of the angles are rays. The rays share a common endpoint (the vertex) Angles are measured in units called degrees.
  • 17. Right Angles Forms a square corner Forms a 90 degree angle. 90 degrees
  • 18. Straight Angle Forms a straight line Angle is 180 degrees 180 degrees
  • 19. Types of Angles When lines intersect to form right angles, then they are classified as perpendicular lines.
  • 20. What do we use to help us? A protractor Here is a standard protractor like you use in the classroom.
  • 21. When we use a protractor, we need to line it up correctly. You need to make sure the protractor is lined up correctly. Is this ready to measure the angle?
  • 22. Were you right.................it wasn’t Look for the upside down ‘T’ in the middle of the straight line on your protractor. This needs to be exactly on the vertex of your angle.
  • 23. We need to remember..... It doesn’t matter which way round the angle is, you ALWAYS need to line the upside down ‘T’ to the vertex of the angle.
  • 24. Now you are ready. Read from the 0°, and follow the inner set of numbers.
  • 25. Homework Page 25 # 1-15 Homework Prentice Hall Geometry
  • 26. QUIZ 1. When two lines are skew, it means three things. What are those three things? 2. Demonstrate how to label a line, plane, ray, and segment. 3. An angle is made up of two ____ sharing a common point called _____.
  • 27. 4. The term skew is a Middle English word meaning "to escape." Skew lines cannot be contained in one plane. Therefore, they have "escaped" a plane. What is something in your life that you have skewed from, for the better?
  • 28. Labeling Angles Must have three points. One on each ray and the vertex. B Labeled as: ∠ BCD or ∠DCB C D
  • 29. Complementary Angles If the sum of the measures of two angles is exactly 90º then the angles are complementary.
  • 30. Supplementary Angles If the sum of the measures of two angles is exactly 180º then the angles are supplementary.
  • 31. Vertical Angles Two angles whose sides are opposite rays.
  • 32. Vertical Angles Vertical angles are congruent Congruent means same or equal. In this picture, <1 and <3 are vertical angles. What is the measure of <2? <2 is 120º because it is congruent to the vertical angle across from it.
  • 33. Adjacent Angles Two coplanar angles with a common side, a common vertex, and no common interior points. ∠ABC is adjacent to ∠CBD A C B D
  • 35. Postulates. Angle Addition Postulate If point B is in the interior of ∠AOC, then m ∠AOB + m ∠BOC = m ∠AOC. A B O C
  • 36. Postulates. Angle Addition Postulate If ∠AOC is a straight angle, then m ∠AOB + m ∠BOC = 180. B A O C