CHAPTER 2 LESSON
4
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"PROOFS
AND MORE COMPLEX FIGURES"
After the quiz, In your book, page 57, "Proofs and More Complex
Figures"
This section
illustrates how we can use the Subtraction Property of Equality to help us in
solving proofs.
RECALL the
Subtraction Property of
Equality: if a = b, then a  c = b  c
Please note in your textbook
on page 57,
what can happen with the Segment Addition Postulate and Angle
Addition Postulate.
Remember that the segment
addition postulate stated that we can add the segments together to create a
larger segment and that the angle addition postulate stated that we can add
adjacent angles together to create a larger angle. Note these two in your book
!!
Given: AB
with C on AB and Given: Ð ACB, with ray CD in the interior
section of Ð
ACB
Now with the Subtraction
Property we can do more than that.
For example:
we can add segments together to create a larger
segment AC + CB = AB
but if we have to isolate a smaller portion
of the line segment
we can subtract
example:
AC = AB  CB or
CB = AB  AC
So this adds to our ability to solve
proofs. The same is true for angles
Note in the book the angle addition
postulate. But if we have to isolate
a smaller angle we can use the subtraction property of equality to do this.
Another example:
Remember the steps that need to be taken to write a proof.
?
What
has to be proven? Then think
of a plan to accomplish this.

1. What do they want you to prove? 2. How are we going to prove this? 
Look at the Given. The m Ð SOQ is equal to the m Ð POR. Now what
angles are a part of Ð SOQ?? Ð 6 and 7. What angles are
a part of Ð POR? Ð 7 and 8.
This is the angle addition postulate because we are
adding Ð
's 6 and 7 together to get Ð SOQ and adding Ð 's7 and 8 to get Ð POR
But we have to get Ð6 to be equal to Ð8.
So let's take the first Ð SOQ
and subtract Ð 7 from it, leaving Ð 6 and then take Ð POR and subtract 7, leaving Ð 8 from it.
Therefore because the given told us that the larger Ð
's SOQ and POR were equal, we can substitute thereby giving us the fact that Ð 6 is equal to Ð 8.
Now let's write the proof. J Please note ***** this information is not required for the
proof. I have supplied it so that you
can see why I used that step.
STATEMENT REASON
1. m Ð SOQ = m Ð POR Given
******(Prove
that you have two angles creating a larger angle)
2. m Ð 6 + m Ð 7
= m Ð SOQ;
m Ð 8 + m Ð 7 =
m Ð
POR Angle
Addition Postulate
********(Now subtract one angle from the larger one
3. m Ð SOQ
 m Ð7 = m Ð6;
m Ð POR
 m Ð 7= m Ð 8 Subtraction
property of equality
4. \ m Ð 6 = m Ð 8 Substitution
\ symbol
for Therefore
Look over Example 1 in your book on page
57. The only above is much like that
one.
Now look at Example 2 on page 58
Again go through the
process to solve a proof. First, What
you do they want you to do?
Answer: Prove
that AB @ CD.
Second, what have they provided you in
the given? That on a line
segment AC @ BD. The problem
is that they have given you a part of the line segment that is not needed,
namely BC.
Third, develop a plan to solve the
proof. What do you know about line
segments? I know that if I
add line segments together that I can create a larger segment. AB + BC = AC
___________________________
A B C AB + BC = AC Angle Addition Postulate
but in this case, I have to get rid of line segment
BC. So I will subtract BC from AC to get
AB and again subtract BC from BD to get CD.
Look at the given again. They
give us that AC @ BD.
SEE BELOW
Then begin to write the proof in a two
categories: Statement and Reason
1. AC @
BD (AC= BD) Given
2. AB + BC = AC; BC + CD = BD Segment Addition Postulate
3. AC  BC = AB; BD  BC = CD Subtraction Property
of Equality
4.\ AB = CD ( AB @ CD)
Substituition
Now look at the last example, #3 on page
58.

First, what do they want you to do? 
Now what do you know about these things? First, perpendicular lines. We know that they
create 90 right angles (Theorem 1.1) You also know that COMPLEMENTARY
ANGLES ARE ALSO 90. Remember
complementary angles are 2 angles equaling 90 degrees.
What do you know about angle bisectors? That they divide the angle into 2 equal
sections.
Now put this to work on the proof.!!!!!!!
Notice that in the given or in the picture that angles 1 and 2 are
complementary. But they want you to
prove that angles 1 and 3 are complementary.
But that is ok, because we know from the given that there is a angle bisector and that means that angles 2 and 3 are
equal to each other. Therefore, we can
switch or substitute 2 and 3 and create the fact that angles 1 and 3 are equal
to 90.
Now you have the plan,
let's write the proof
1. Ray BA ^
Ray BC Given
2. m Ð
1 + m Ð 2 = 90 If
outer rays of acute adjacent angles are
perpendicular,
then the sum of the angles is 90
3. Ray BC bisects Ð DBE Given
4. m Ð
3 = m Ð 2 Definition
of angle bisector
5. \ m Ð 1 +
m Ð 3 = 90 Substitution
Just a
couple of things
When you submit your answers to the
problems, you are not going to be able to use a symbol for the word (example
perpendicular ^ ) you will have to write out the word in your
statement.
For example in number 1 above, you would
have to write
1. Ray BA is perpendicular to Ray BC.
Now
let's begin to work. Page 6061. Work on problems 414 even only.
Student's First & Last Name:
Please number your answers
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