Theorems Relating Lines and
Planes
Review From Yesterday
• What is the difference between collinear points and
coplanar lines?
– Answer: collinear points are points on the same
line and coplanar lines are lines in the same plane
• What happens when two planes intersect?
- Answer: ONE line is formed
• What are the 4 ways you can create a plane?
- Answer: 3 noncollinear points, a line and a point off
the line, two intersecting lines, two parallel lines
Theorem 1
• If a line is perpendicular to each of two
intersecting lines at their point of
intersection, then the line is perpendicular to
the plane determined by them.
Theorem 2
• Through a given point there passes one and
only one plane perpendicular to a given
line.
Theorem 3
• Through a given point there passes one and
only one line perpendicular to a given
plane.
Theorem 4
• If two planes are perpendicular to the same
line, they are parallel.
Let’s work on Some Questions
• A line perpendicular to a plane is
perpendicular to every line in the plane.
True of False?
• Answer: False
• Through a point outside a plane, one
and only one plane can be drawn
perpendicular to the plane. True or False?
Answer: False
More Practice!
• Fill in the blank: Two planes that are
perpendicular to the same line are
_____________ to each other
– Answer: parallel
• Two lines _____________ to the same
plane are parallel.
– Answer: Perpendicular
3
4
4
2
4
1. Line k is drawn so that it is perpendicular to two distinct planes, P and R.
What must be true about planes P and R?
(2) Planes P and R are parallel.
2. Lines and intersect at point E. Line m is perpendicular to lines k 1 and k2
at point E.
Which statement is always true?
(3) Line m is perpendicular to the plane
3. For this rectangular solid, Planes EGH and GAB intersect to form line
GH
4. One dihedral angle formed by planes l and q is:
P – YZ – A
5. If lines n and m are each perpendicular to plane A, what is the relationship
between n and m ?
n and m are coplanar
6. What happens when two planes intersect? ________________________________
One line is formed