Download Chapter 2: Basic Concepts and Proof 2.1 perpendicularity

notes 2.4
September 14, 2016
Chapter 2: Basic Concepts and Proof
Chapter 2 - Openers and notes
Resources (1) Theorem/postulate packet
2.1 perpendicularity
(2) Vocabulary - download from website
(3) Answer key to chapter 2 Review exercises
(4) Homework
2.1
Describe how to find the shortest distance from a point to a line?
In algebra you studied relationships between points when you
graphed on a coordinate system. How are the axes related on
the coordinate plane?
In the diagram,
name the right angles
G
E
F
H
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E
A
1 23
C
D
4
B
Angles 1,2,3 and 4 are
in the ratio of 2:1:1:2,
find the measure of
<EBC
F
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C
A
GIVEN: <ACB=90,
B
D
Prove: <C
<D
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Chapter 2: Basic Concepts and Proof
2.1 perpendicularity
Find the area of rectangle
ABCD
Find coordinates of D
If the coordinates of B are
not given, could you
answer the first two
questions?
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Explain perpendicularity:
What would be a good definition?
What does the word oblique mean?
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GIven: a τ b
Prove: < 1 ≅ < 2
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A ( 3, 2)
Where is A ' if it is a
result of rotating A 90
degrees about the
origin?
WHat is the result of
rotating B 90 degrees
about the origin if B
is at (0, -3)?
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LIne a is perpendicular to line b. The angle
formed by their intersection is trisected. One
of the new angles then is also trisected. One
of these newer angles is bisected. How large is
the smallest angle?
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What is the relationship algebraically
between perpendicular lines?
Can you find the equation of
the perpendicular line through
B to side AC ?
B(7, 5)
A(1, 1)
C(9, 3)
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2.2
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Complementary and Supplementary Angles
A (-2,8)
(1,8) B
what are the
coordinates of D ?
x
C
(-2,-3)
y
( , )D
What is the area and
perimeter of the
rectangle
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If two angles are complementary then
________________________________________
________________________________________
If two angles are supplementary then
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Find the complement and the
supplement of
The supplement of an angle is five times the complement
of the angle. Find the measures of the original angle,
the complement and the supplement:
let x be the angle
180 - x is the supplement
?
is the complement. Now translate the problem
into an equation
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A
C
S
T
given: <CAS and <TAS are
complementary
Prove:
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2.3 Drawing Conclusions
It is important as a student of Geometry to
analyze and hypothesize and draw conclusions
based on your observations, theorems, postulates
and definitions. Be CAREFUL to not ASSUME
!
Can you draw a conclusion based on the information?
given: <1 <2
conclusion:
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Q
Given: M is the midpoint of GH
Conclusion:
G
H
M
P
Given: <1
Conclusion:
<2
<3
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Our task is to solve the problem, but before we just jump in...let us
analyze. We can use definitions, theorems & postulates and then
draw conclusions.
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Once again, can we draw conclusions based on the information given?
Let's proceed...
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E
A
What type of 'assumptions' or
conclusions can we make?
C
M
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2.4 Congruent Supplements
and congruent Complements
suppose <1 is supplementary to <2
and also suppose that <2 is
supplementary to <3, what
conclusion can you draw?
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Now suppose that <1 is supplementary to <2 an
<3 is supplementary to <4 and also that <2 <4,
what conclusion can you draw?
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suppose that <1 is complementary to <2 and
that <3 is complementary to <2, what conclusion can be
drawn?
Suppose that <1 is complementary to <2
<3 is complementary to <4 and also that <1
What conclusion can be drawn?
<3
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theorems:
If two angles are supplementary to the same angle then they
are congruent
If two angles are supplementary to congruent angles then
they are congruent
If two angles are complementary to the same angle then they
are congruent
If two angles are complementary to congruent angles then
they are congruent
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Given: diagram as
shown
Prove <1
<3
now, do #1-8 in BIG BLUE and #19
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2.5 Addition and Subtraction
Properties
A
B
M
C
Let AB =CM , can you draw any
conclusions?
W
N
S
T
I
Let <WIN <SIT,
can you draw any conclusions?
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S
Can you conclude anything
about <SNL and <SLN?
E
N
L
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Addition Properties for segments and angles
theorems 8-11
Subtraction Properties for segments and angles
theorems 12-13
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A
B
C
D
Given <BAD
<CAE
Prove: <BAC
E
F
<FAD, AD bisects
<FAE
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2.6 Multiplication and Division Properties
M
N
O
P
Q
R
S
Let the points N and O be trisection points AND
R and S be trisection points.
ALSO, MN=QR. can you conclude anything?
T
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W
Y
A
P
E
X
T
Z
Ray XY and Ray ZW are angle bisectors, if
<1 = <3, what can be said of <AXE and
<PZT ?
Multiplication and Division Theorems 14 and 15
Using the theorems 14 & 15
1) Look for double use of the words bisect or trisect of midpoint
2) use the Mult. Prop. if segments or angles are greater in the
prove statement
3) use the Div Prop. if the segments or angles are smaller in the
prove statement.
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2.7 transitive and
Substitution properties
If <A =<B and <B = <C, is <A = <C ?
Does this sound like a familiar property?
Theorems 16 & 17
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P
Q
S
G: Q and R are
midpoints and PS=PT
P: QS = RT
R
T
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2.8 Vertical Angles
Name the vertical angle pairs
E
H
G
D
B
J
C
K
A
Which angle is a vertical angle to <GBK ?
to <EBA ?
to <ABG ?
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G: GD bisects <CBE
P: <1
<2
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If a pair of vertical angles are complementary, what are their measures?
If a pair of vertical angles are supplementary, what are their measures?