MORE Theorems Relating
Lines and Planes
Theorem 5
• Two lines perpendicular to the same plane
are coplanar.
Given : AB  planeq, CD  planeq
Conclusion: AB and CD are in planel (in the sam e plane)
Theorem 6
• If a plane intersects two parallel planes,
then the intersection is two parallel lines.
Given : plane l
plane r ,
Conclusion: AB
CD .
Parallel lines are formed!
Theorem 7
• If a line is perpendicular to a plane, then
every plane containing the line is
perpendicular to the given plane.
Given : AB  plane l
Conclusion : plane l  plane q
Theorem 8
• Two planes are perpendicular to each other
if and only if one plane contains a line
perpendicular to the second plane.
planeq  planel , if and onlyif
AB is in planeq and AB  planel
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
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!
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
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) 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
4
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) 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
Let’s work on Some Questions
1. If plane A is perpendicular to a line in plane B
then planes A and B are perpendicular to each
other.
- Answer: True
2. Line r and line s are perpendicular to plane K,
what can be said about lines r and s?
- 2) they are coplanar
3. 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
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
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