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Relating lines to planes Lesson 6.1

This lesson explores the relationship between lines and planes in geometry, focusing on defining coplanar and non-coplanar points, as well as the criteria for determining a plane. Four methods to establish a plane using points and lines are discussed, alongside important postulates and theorems that guide geometric reasoning. Students will learn to identify the intersection points and apply these principles to prove geometric relationships, enhancing their understanding of two-dimensional space without thickness or edges.

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Relating lines to planes Lesson 6.1

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  1. Relating lines to planesLesson 6.1 Dedicated To Graham Millerwise

  2. Plane • Two dimensions (length and width) • No thickness • Does not end or have edges • Labeled with lower case letter in one corner m

  3. A B m C Coplanar Points, lines or segments that lie on a plane B A C m Non-Coplanar Points, lines or segments that do not lie in the same plane

  4. A B m C Definition: Point of intersection of a line and a plane is called the foot of the line. B is the foot of AC in the plane m.

  5. 4 ways to determine a plane 1. Three non-collinear points determine a plane. One point - many planes Two points - one line or many planes Three linear points - many planes n

  6. 2. Theorem 45: A line and a point not on the line determine a plane.

  7. 3. Theorem 46: Two intersecting lines determine a plane.

  8. 4. Theorem 47: Two parallel lines determine a plane.

  9. Two postulates concerning lines and planes Postulate 1: If a line intersects a plane not containing it, then the intersection is exactly one point. X C m Y

  10. m Postulate 2: If two planes intersect, their intersection is exactly one line. n

  11. mՈn = ___ • A, B, and V determine plane ___ • Name the foot of RS in m. • AB and RS determine plane ____. • AB and point ______ determine plane n. • Does W line in plane n? • Line AB and line ____ determine plane m. • A, B, V, and _______ are coplanar points. • A, B, V, and ______ are noncoplanar points. AB m P n R or S No VW W or P R or S

  12. Given: ABC lie in plane m PB  AB PB  BC AB  BC Prove: <APB  <CPB P B A C m • PB  AB, PB  BC • PBA & PBC are rt s • PBA  PBC • AB BC • PB  PB • ΔPBA  ΔPBC • APB  CPB • Given •  lines form rt s • Rt s are  • Given • Reflexive Property • SAS (4, 3, 5) • CPCTC

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