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Geometry Notes, Chapter 1-2
In these sections, if you know the vocabulary, many questions are common sense. Use your text
to review any vocab listed below that you don’t know in each section. These notes are list of
material that you are responsible for in each section. If I make changes from the book, those
changes are specified here.
You NEVER have to memorize Postulate or Theorem numbers (eg. Theorem 1-2, etc.)
If you forget the name of a postulate or theorem, it is always OK to state the theorem or postulate.
Section 1-2:
Section 1-3:
vocab :
vocab:
point
plane
space
coplanar points
bisector
line
undefined terms
collinear points
intersection
segment
opposite rays
coordinate
postulate
congruent segments
ray
number line
length
congruent
rules:
Ruler Postulate (just understand concept)
Segment Addition Postulate (know inside out)
• can be abbreviated to SAP or Seg.
Add Post
• can only be used to add TWO
segments at a time!
Section 1-4:
Section 1-5:
vocab :
Postulate 6 : “Through any two points there
exists exactly one line” - used to justify adding
a line to a diagram by connecting two points
angle
vertex
right angle
straight angle
adjacent angles
sides of an angle
acute angle
obtuse angle
congruent angles
bisector of an angle
rules :
Protractor Postulate (understand concept)
Angle Addition Postulate
• know both parts completely!!
• can be abbreviated to AAP or
∠ Add. Post.
• can only be used to add TWO angles
at a time!
The other postulates are rules that you should
be able to visualize, but not necessarily "use"
in a problem.
Section 2-1:
vocab :
if-then statements (a.k.a. conditional statements or conditionals)
hypothesis
conclusion
counter-example
biconditional (if and only if) : true only if both conditionals are true
(“If and only if” is often abbreviated to “iff” by mathematicians but read as “if and only if”)
Can definitions always be written as biconditionals? Answer: Yes
Can postulates and theorems always be written as biconditionals? Answer: No, only
sometimes… keep this in mind as you learn new theorems throughout the year
Section 2-2:
With the following properties, you should know both the technical definitions written in your text and the
translations (even more important) below. These should be memorized.
addition property : "you can add equal things to both sides of an equation"
subtraction property : "you can subtract equal things from both sides of an equation"
multiplication property : "you can multiply both sides of an equation by the same number"
division property : "you can divide both sides of an equation by the same number (except 0)"
substitution property : "the eraser principal : in an equation or inequality, you may erase one quantity and
replace it by a quantity which has been shown to be equal"
reflexive property : "anything equals itself"
symmetric property : "you can switch what is on the left and right side of the = sign"
transitive property : "if one thing equals a second and a second equals a third, then the first equals the
third" (You may extend this further: if a=b and b=c and c=d … and y=z, then a=z)
distributive property (as defined in the text, familiar to you as factoring out a common term)
Questions to ask yourself: If you use substitution, ask “What is the original equation? What thing is being
replaced by an equal thing?” If you use something like the addition property, ask yourself “What is the
original equation? What equal things are being added to both sides of the equation?”
For substitution, you MUST refer to the original equation and the equation/s that you use to substitute
from. Example: substitution (5,3,2) where statement 5 is the original equation.
For addition and subtraction, also refer to the original equation and the line of the equality that gets added
or subtracted to both sides. Note: you cannot add or take away the same thing from both sides, just equal
things. You may need the reflexive property to make equal. One exception: you may add or subtract the
same number to both sides of the equation.
addition of = (5, 7)
On pp, 38-39, you should read the 3 examples, and see how each reason justifies each statement.
Review the Segment Addition Postulate and the Angle Addition Postulate. MEMORIZE exactly
what each postulate justifies. Note that the Angle Addition Postulate justifies two different rules.
Section 2-3:
Midpoint Theorem (Midpt Thm): know the theorem, how to use it in a proof, and understand the
proof of the theorem (in text). Recall that definitions are biconditionals!
Angle Bisector Theorem ( ∠ Bis Thm): know the theorem and understand the proof
Know the difference between the Midpoint Theorem and the Midpoint Definition!!!: the
definition says that the point splits the segment into two equal parts and can be used like a
biconditional (if it's a midpoint then the two segment are equal; if the two segments are equal,
then it's a midpoint). The theorem only deals with doubling and halving.
Likewise, know the difference between the Angle Bisector Theorem and Definition (just like the
midpoint ones).
Note: when referring to theorems, NEVER refer to the theorem number in the text; either call it
by the name in the text (if there is one), or write it out in words. In our book, VAT is Theorem 23 because it’s the third theorem in chapter 2. In another book, it might be the 4th theorem in
chapter 5. The theorem numbers have no real meaning so it’s not worth knowing them!
Section 2-4:
Section 2-5:
vocab :
vocab :
complementary angles (complement)
supplementary angles (supplement)
vertical angles
Perpendicular lines (know the 2 conclusions
the definition justifies - see p.56) and
remember definitions are biconditionals!
rule :
Vertical Angle Theorem (a.k.a. VAT) - know
the theorem and how to prove it!
Know the three theorems on page 56. You
must write these out in words. Using “if” and
“then” can be helpful. Using the diagrams on
page 56, examples are :
Section 2-6:
if line JK is perp. to line MN, then angle 1 is
congruent to angle 2 (reason: if 2 lines are
perp, they form congr, adjacent angles)
Know the two theorems, 2-7 (called The
Supplementary Angle Theorem) and 2-8 (The
Complementary Angle Theorem), and be able
to prove them
(the proof of 2-7 is in the text)
if angle 1 is congruent to angle 2, then the
lines are perpendicular (reason: if 2 lines form
congr, adjacent angles, then the lines are perp)
if AO is perp. to OC, then angle AOB and
angle BOC are complementary (reason: if the
ext sides of 2 adjacent angles are perp, then the
angles are complementary)
The symbol ⊥ can always be used in place of writing the word “perpendicular” (eg. “AB is
perpendicular to BC” can be written as simply “AB ⊥ BC”)
Making Geometry flashcards (optional)
If flashcards help you remember rules better, the best way to do this for many rules is to draw a
diagram of a diagram with the “if” on one side (in pictures) and the “then” along with the rule on
the other side.
Examples:
Side 1 of card (the if part of a rule)
side 2 of card (the then part of the rule)
m∠1+ m∠2=180
by AAP (Angle Addition Postulate)
1
2
The lines are perpendicular by the theorem
that says: if 2 lines form congruent adjacent
∠ ’s, then ⊥
1
B
2
C
A
A is the midpt of BC by the Def of
Midpt
(midpoint: a point that breaks a segment
into 2 congruent segments)
A is the midpoint of BC
B
A
AB = BC by Def of Midpt.
C
(midpoint: a point that breaks a
segment into 2 congruent
segments)
AB =
1
BC by Midpt. Thm
2
or
AB = AC by Midpt. Def
A is the midpoint of BC
B
A
C
Some rules don’t get triggered by a diagram, so you might do this:
Substitution property
If you have an equation, then you can replace
one thing with something known to be equal to
it. Be sure to refer to the original equation, and
the equations that show the equal things being
switched, like:
Substitution (5, 3)
addition property of equality
If you have an equation, you may add equal
things to both sides.
Hint: identify the original equation, and the
equals that are added to both sides