Lesson 10.3
Perpendiculars in Space
pp. 423-427
Objectives:
1. To define perpendicular figures in
space.
2. To prove theorems involving
perpendiculars in space.
Definition
Perpendicular planes are two
planes that form right dihedral
angles.
D
B
C
E
A
D-AB-E is a right dihedral
angle.
Definition
A line perpendicular to a plane
is a line that intersects the
plane and is perpendicular to
every line in the plane that
passes through the point of
intersection.
Definition
A perpendicular bisecting
plane of a segment is a plane
that bisects a segment and is
perpendicular to the line
containing the segment.
x
x
Theorem 10.2
A line perpendicular to two
intersecting lines in a plane is
perpendicular to the plane
containing them.
l
m
p
A
n
Theorem 10.3
If a plane contains a line
perpendicular to another
plane, then the planes are
perpendicular.
n
A
D
E
B
m
C
Theorem 10.4
If intersecting planes are each
perpendicular to a third plane,
then the line of intersection of
the first two is perpendicular to
the third plane.
Theorem 10.5
If AB is perpendicular to plane
p at B, and BC  BD in plane p,
then AC  AD.
Theorem 10.5
A
p
B
C
D
Theorem 10.6
Every point in the
perpendicular bisecting plane
of segment AB is equidistant
from A and B.
Theorem 10.7
The perpendicular is the
shortest segment from a point
to a plane.
Two planes perpendicular to
the same plane are parallel?
1. True
2. False
Two lines perpendicular to the
same plane are parallel?
1. True
2. False
Two planes perpendicular to
the same line are parallel?
1. True
2. False
Homework
p. 427
►A. Exercises
Make a sketch of the following.
1. A dihedral linear pair
►A. Exercises
Make a sketch of the following.
3. Theorem 10.5 A
p
B
C
D
►A. Exercises
Make a sketch of the following.
5. Theorem 10.7
►A. Exercises
Use the diagram for exercises 6-7.
7. State an Angle Addition Theorem for
Dihedral Angles.
B
C
•
•
• XD
A
►A. Exercises
Explain or define each term below.
9. Complementary dihedral angles
►B. Exercises
Draw conclusions about dihedral angles
(based on your knowledge of plane
angles).
11. Vertical dihedral angles are . . .
►B. Exercises
Draw conclusions about dihedral angles
(based on your knowledge of plane
angles).
13. Supplementary dihedral angles that
are adjacent . . .
►B. Exercises
Draw conclusions about dihedral angles
(based on your knowledge of plane
angles).
15. If one dihedral angle of a dihedral
linear pair is a right angle, then . . .
►C. Exercises
Prove each theorem below.
16. Theorem 10.4
Given: m ⊥ p, n ⊥ p,
m intersects n
Prove: AB ⊥ p
m A
E
B
F D
n
C
p
Statements
Reasons
1. m  p, n  p, & 1. Given
m intersects n
2. m  n = AB
2. Plane Intersection
Postulate
3. A-BC-D and 3. Definition of
A-BE-F are
perpendicular
right angles
planes
Statements
Reasons
4. mA-BC-D=90 4. Def. of right
mA-BE-F=90
dihedral angles
5. mA-BC-D=
5. Def. of dihedral
mABD
angle measure
mA-BE-F=
mABF
6. mABD=90
6. Transitive prop.
mABF=90
of equality
Statements
Reasons
7. ABD & ABF 7. Def. of right
are right angles
angle
8. AB  DB
8. Def. of perpend.
AB  BF
lines
9. AB  p
9. Line perpend. to
two intersecting
lines is perpend.
to the plane
containing them
►C. Exercises
Prove each theorem below.
17. Theorem 10.5
Given: AB ⊥ p at B,
A
BC ≈ BD
C, D  p
Prove: AC  AD
B
C
p
D
■ Cumulative Review
19. Name two postulates for
proving triangles congruent.
■ Cumulative Review
20. Name two theorems for
proving triangles congruent.
■ Cumulative Review
21. Name a fifth method for
proving triangles congruent
that works only for right
triangles.
■ Cumulative Review
22. What is SAS called when
applied to right triangles?
■ Cumulative Review
23. What is ASA called when
applied to right triangles?