Theorems Relating Lines and Planes

Theorems Relating Lines and Planes

Review From Yesterday• What is the difference between collinear points and

coplanar lines?

• What happens when two planes intersect?

• What are the 4 ways you can create a plane?

– Answer: collinear points are points on the same line and coplanar lines are lines in the same plane

- Answer: ONE line is formed

- 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

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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 and

at point E.

Which statement is always true?

k1k2

(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 ?

6. What happens when two planes intersect? ________________________________

n and m are coplanar

One line is formed