Transcript 2.4 Use Postulates & Diagrams

2.4 Use Postulates & Diagrams
Objectives:
1. To illustrate and understand postulates
about lines and planes
2. To accurately interpret geometric
diagrams
3. To use properties of special pairs of
angles to find angle measurements
Example 1
What is the length of SM ?
S
A
6 cm
20 cm
M
Example 1
You basically used the Segment Addition
Postulate to get the length of the segment,
where SA + AM = SM.
S
A
26 cm
M
Postulates
As you build a
deductive system
like geometry, you
demonstrate that
certain statements
are logical
consequences of
other previously
accepted or proven
statements.
Postulates
This chain of logical
reasoning must
begin somewhere,
so every deductive
system must contain
some statements
that are never
proved. In
geometry, these are
called postulates.
Postulates and Theorems
• Postulates are
statements in
geometry that are so
basic, they are
assumed to be true
without proof.
– Sometimes called
axioms.
• Theorems are
statements that were
once conjectures but
have since been
proven to be true
based on postulates,
definitions, properties,
or previously proven
conjectures.
Both postulates and theorems are ordinarily written in conditional form.
(If-Then statement)
Postulates Are Easy!
For example, under the second strip write:
Through any two points there exists
exactly one line.
B
A
B
A
Seven More!
Example 1
State the postulate illustrated by the
diagram.
2 lines intersect at
exactly one point
2 planes intersect
at a line
Example 2
How does the
diagram shown
illustrate one or
more
postulates?
Answer in your
notebook
Interpreting Diagrams
When you interpret a
diagram, you can
assume only
information about
size or measure if it
is marked.
Don’t make assumptions
Interpreting Diagrams
Interpreting Diagrams
Example 3
Sketch and carefully label a diagram with
plane A containing noncollinear points R,
O, and W, and plane B containing
noncollinear points N, W, and R.
Do this in your notebook
Perpendicular Figures
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.
Example 4
Which of the following
cannot be assumed
from the diagram?
1. A, B, and F are
collinear.
2. E, B, and D are
collinear.
3. AB  plane S
2
Example 4
Which of the following
cannot be assumed
from the diagram?
4. CD  plane T
5. AF intersects BC at
point B.
4
Example 5a
1. Identify all
linear pairs of
angles.
2. Identify all
pairs of vertical
angles.
Answer in notebook
2
3
1
4
Example 5b
3. If m<1 = 40°,
find the
measures of
the other
angles in the
diagram.
2
3
1
Answer in you notebook
4
Linear Pair Postulate
If two angles form a linear pair, then they are
supplementary.
Do we have to prove this?
Vertical Angle Congruence Theorem
Vertical angles are congruent.
Example 6
Find the missing measure of each angle.
60
A
D
B
C
65
<A=65, <B=55, <C=60, <D=55
Example 7
Find the value of x and y.
3y - 1
x=70, y=12
2x +5
35
Example 8
Find the value(s) of x.
x=1/2, 7
½ does not create congruent
angles, therefore the only
reasonable answer is 7
Example 9: SAT
For the two intersecting lines, which of the
following must be true?
I. a > c
II. a = 2b
III. a + 60 = b + c
lll
a
60
b
c
Example 10: SAT
In the figure, what is
the value of y?
x
3x
20
y
2x