GEOMETRY
2.3 Postulates and Diagrams
Decide whether inductive or deductive reasoning is used to reach the conclusion.
1. Everyday when you come home from soccer practice, your dog, Max, greets you at the
door. Thus, when you get home from practice today, Max will be waiting for you at the
door.
Determine if it is possible to use the Law of Detachment. If yes, what can you
conclude from the given information?
2. If two segments have the same measure, then the segments are congruent
segments.
AC = BD
3. If you are sick, then you make an appointment to see the doctor. You have an
appointment to see the doctor.
Determine if it is possible to use the Law of Syllogism. If yes, write a new conditional
statement that follows from the pair of true statements.
4. If you paint your house, then it will look nice. If your house looks nice, then someone
Septemberbuy
25, 2015
2.3 POSTULATES AND DIAGRAMS
will
it.
ESSENTIAL QUESTION
In a diagram, what can be assumed and what needs to be
labeled?
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2.3 POSTULATES AND DIAGRAMS
REVIEW
What is a Postulate?
A rule that is accepted without proof.
Section 2.3 introduces seven more postulates to us.
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2.3 POSTULATES AND DIAGRAMS
THE POSTULATES
2.1 Two Point Postulate
Through any two points, there
A
exists exactly one line.
Given points A & B, the only line that can be
drawn through them is line l.
B
l
2.2 Line-Point Postulate
A line contains at least two points.
Given line l, there exists at least two points, A & B, on
the line.
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2.3 POSTULATES AND DIAGRAMS
THE POSTULATES
m
2.3 Line Intersection Postulate
If two lines intersect, then their
intersection is exactly one point.
C
Lines m and n intersect at point C.
n
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2.3 POSTULATES AND DIAGRAMS
THE POSTULATES
2.4 Three Point Postulate
Through any three noncollinear points,
there exists exactly one plane.
Given noncollinear points D, E, & F, plane
R is the only plane containing them.
R
D
2.5 Plane-Point Postulate
E
F
A plane contains at least three
noncollinear points.
Given plane R, there are three noncollinear points D, E, & F,
in the plane.
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2.3 POSTULATES AND DIAGRAMS
THE POSTULATES
2.6 Plane-Line Postulate
R
If two points lie in a plane,
then the line containing them
lies in the plane.
Points D and E lie in plane R,
so line DE is also in plane R.
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D
E
F
2.3 POSTULATES AND DIAGRAMS
THE POSTULATES
S
2.7 Plane Intersection Postulate
T
If two planes intersect, then
their intersection is a line.
Plane S and plane T intersect at
line l.
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l
2.3 POSTULATES AND DIAGRAMS
EXAMPLE 1
Identify the postulate illustrated below.
a.
If
b.
, then
2.3 Line Intersection Postulate
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.
If
, then
.
2.7 Plane Intersection Postulate
2.3 POSTULATES AND DIAGRAMS
EXAMPLE 2A
Write an example of the Plane-Point Postulate and the Plane-Line Postulate.
Plane-Point Postulate
Plane P contains at least three
noncollinear points, A, B, and C.
Plane-Line Postulate
m
Q
C
B
A
n
Point A and point B lie in plane P.
So, line n containing points A and
B Also lies in plane P.
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2.3 POSTULATES AND DIAGRAMS
EXAMPLE 2B
Which postulate allows you to say that the intersection of plane P and plane Q
is a line, line m?
2.7 Plane Intersection
m
Q
C
Postulate
B
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A
n
2.3 POSTULATES AND DIAGRAMS
EXAMPLE 3
Sketch a diagram showing 𝑇𝑉 intersecting 𝑃𝑄 at point W, so that
𝑇𝑊 ≅ 𝑊𝑉.
P
W
V
T
Q
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2.3 POSTULATES AND DIAGRAMS
LINE PERPENDICULAR TO A PLANE
A line is perpendicular to a plane if
and only if the line intersects the
plane in a point and is
perpendicular to every line in the
plane that intersects it at that point.
t
A
In a diagram, a line perpendicular
to a plane must be marked with a
right angle symbol as shown.
C
p
q
Line t is perpendicular to lines p and q, so
line t is perpendicular to plane A.
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2.3 POSTULATES AND DIAGRAMS
EXAMPLE 4
T
A
Which of the following statements
cannot be assumed from the diagram?
Points A, B, and F are collinear.
S
E
C
B
D
Points E, B and D are collinear.
𝐴𝐵 ⊥ 𝑝𝑙𝑎𝑛𝑒 𝑆
F
𝐶𝐷 ⊥ 𝑝𝑙𝑎𝑛𝑒 𝑇
𝐴𝐹 intersects 𝐵𝐶 at point B.
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2.3 POSTULATES AND DIAGRAMS
EXAMPLE 5
T
A
a. Can you assume that plane
S intersects plane T at 𝐵𝐶?
Yes!
E
b. Explain how you know that
𝐴𝐵 ⊥ 𝐵𝐶.
𝐴𝐵is perpendicular to plane S,
and 𝐵𝐶 is on plane S and
intersects 𝐴𝐵 at B.
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S
C
B
D
F
2.3 POSTULATES AND DIAGRAMS
ASSIGNMENT
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2.3 POSTULATES AND DIAGRAMS