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MORE Theorems Relating Lines and Planes

MORE Theorems Relating Lines and Planes. Theorem 5. Two lines perpendicular to the same plane are coplanar. Theorem 6. If a plane intersects two parallel planes, then the intersection is two parallel lines. Parallel lines are formed!. Theorem 7.

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MORE Theorems Relating Lines and Planes

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  1. MORE Theorems Relating Lines and Planes

  2. Theorem 5 • Two lines perpendicular to the same plane are coplanar.

  3. Theorem 6 • If a plane intersects two parallel planes, then the intersection is two parallel lines. Parallel lines are formed!

  4. Theorem 7 • If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.

  5. Theorem 8 • Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.

  6. The Dreaded Question:Infinite vs. One?? • Most questions that ask how many planes are perpendicular to another plane will have INFINITE as the answer! Ex: Line m is perpendicular to plane A. How many planes can contain line m and also be perpendicular to plane A? (1) 1 (2) 2 (3) 0 (4) infinite 4

  7. The Dreaded Question:Infinite vs. One?? • Most questions that ask: • How many lines are perpendicular to a plane through any point • How many planes are perpendicular to a line and pass through a point on the line Will have ONE as the answer!

  8. Example of “one” • Point P is on line m. What is the total number of planes that are perpendicular to line m and pass through point P? (1) 1 (2) 2 (3) 0 (4) infinite 1

  9. Mixed Practice 1. Given plane A and point C which is not on plane A. How many lines can be drawn through point C that are perpendicular to plane A? (1) 1 (2) 2 (3) 0 (4) infinite 2. Given plane R and point S not on plane R. How many planes can be drawn perpendicular to plane R and pass through point S? (1) 1 (2) 2 (3) 0 (4) infinite 1 4

  10. Mixed Practice 3. Line segment JK is in plane r and point A is on line segment JK. How many planes can pass through point A that are perpendicular to JK? (1) 1 (2) 2 (3) 0 (4) infinite 4. Point T is not on plane m. How many lines pass through point T and are perpendicular to plane m? (1) 1 (2) 2 (3) 0 (4) infinite 1 1

  11. Let’s work on Some Questions • If plane A is perpendicular to a line in plane B then planes A and B are perpendicular to each other. - Answer: True • Line r and line s are perpendicular to plane K, what can be said about lines r and s? - 2) they are coplanar • Given planes A and B. Line m is in plane A. Line m is perpendicular to plane B. What is the relationship between planes A and B? - Answer: 3) Planes A and B are perpendicular

  12. More Plane Review 4. Given line segment AB in plane m and point C, on line AB. How many planes pass through point C that are perpendicular to AB? • Answer: 1 5. Through a point outside the plane, only one line can be drawn perpendicular to the plane. True or False? • Answer: True

  13. More Practice 6. If lines k and s are both perpendicular to plane C, then: - Answer: 2) Lines k and s are coplanar and parallel to each other 7. Plane T and plane S are parallel to each other and plane Q intersects both plane T and plane S. What is formed by both intersections? • Answer: parallel lines

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