1. MORE Theorems Relating Lines and Planes

2. Two lines perpendicular to the same plane are coplanar.
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