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1-5 – Postulates and Theorems Relating Points, Lines, and Planes

1-5 – Postulates and Theorems Relating Points, Lines, and Planes. Terms. ________________________ – a basic assumption accepted without proof ________________________ – a statement that can be proved using postulates, definitions, and previously proved theorems

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1-5 – Postulates and Theorems Relating Points, Lines, and Planes

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  1. 1-5 – Postulates and Theorems Relating Points, Lines, and Planes

  2. Terms • ________________________ – a basic assumption accepted without proof • ________________________ – a statement that can be proved using postulates, definitions, and previously proved theorems • ________________________ – there is at least one • ________________________ – there is no more than one • ________________________ – exactly one • ________________________ – to define or specify

  3. Postulates • What Postulates have we discussed so far?

  4. Postulates Postulate 5 – a line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane.

  5. Postulates Postulate 6 – Through any two points there is exactly one line.

  6. Postulates Postulate 7 – Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.

  7. Postulates Postulate 8 – If two points are in a plane, then the line that contains the points is in that plane.

  8. Postulates Postulate 9 – If two planes intersect, then their intersection is a line.

  9. Theorems Theorem 1-1 – If two lines intersect, then they intersect in exactly one point.

  10. Theorems Theorem 1-2 – Through a line and a point not in the line there is exactly one plane.

  11. Theorems Theorem 1-3 – If two lines intersect, then exactly one plane contains the lines.

  12. Name the Postulate/Theorem • Relationships between points • Two points must be collinear • Three points may be collinear or noncollinear • Three points must be coplanar • Three noncollinear points determine a plane • Four points may be coplanar or noncoplanar • Four noncoplanar points determine space • Space contains at least four noncoplanar points

  13. Name the Postulate/Theorem • Three ways to determine a plane • Three noncollinear points determine a plane • A line and a point not on the line determine a plane • Two intersecting lines determine a plane

  14. Relationships between two lines in the same plane Either two lines are parallel, or they intersect. Example:

  15. Relationship between a line and a plane Either a line and a plane are parallel, or they intersect in exactly one point, or the plane contains the line. Example:

  16. Relationships between two planes Either two planes are parallel, or they intersect in a line. Examples:

  17. Practice • Name a plane that contains . • Name a plane that contains but is not shown in the diagram. • Name a plane that contains and . • Name a plane that contains and . W V T U S R P Q

  18. Practice • Name the intersection of plane TWRQ and plane PSWT. • Name five lines in the diagram that don’t intersect plane UVRQ. • Name one line that is not shown in the diagram that does not intersect plane UVRQ. • Name three planes that don’t intersect and don’t contain . W V T U S R P Q

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