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Geometry Journal 2
Nicolle Busto 9-1
9-1
Conditional Statement
• It is a statement that establishes a necessary
condition for a thing to happen.
Examples:
1. If it rains then the grass will be wet.
2. Converse: If the grass is wet then it rained.
3. Inverse: If it doesn’t rain then the grass will not be wet.
4. Contrapositive: If the grass is not wet then it didn’t rain.
Counter example
• It’s an example that
proves that a statement
is false.
Example:
1. If the grass is wet then
it rained.
-The grass could be wet, if
you watered the grass.
2. If a number is divisible by
5 then its unit digit is 5.
-20
3. If a number is odd
then the number is
prime.
-9
Definition
A description of a
mathematical concept
that can be written as a
bi-conditional statement.
Perpendicular lines: Two
lines are perpendicular iff
they intersect at 90°.
A line is perpendicular to a
plane iff their intersection
is exactly a 90° angle.
Bi-Conditional statement
It is a statement where both
conditions are needed.
They are used to write
definitions
They are important because
they can be used to
establish if a definition is
true or false.
Example 1: A rectangle is a
square iff their four sides
are equal.
Example 2: An angle is acute
iff its measure is less than
90°.
Example 3: A triangle is
equilateral iff its three sides
are congruent.
Deductive reasoning
Examples: p = today is a
school day. Q = I am
going to the school
today.
-PQ
2. P = today the ice
cream store is open.
Q = I am going to
the store today.
- PQ
3. P = If two numbers
are odd. Q = their sum
is even.
Is the process of using
logic to draw conclusions.
Symbolic notation: Uses
letters to represent
statements and symbols
to represent the
connections between
them. It works by reaching
conclusions from a
starting statement.
Laws of Logic
1. Law of Detachment: If P  Q is a true statement and P
is true then Q is true.
·Example 1: If an animal is a dog  it has a tail.
- The pet of Busto family is a dog  the pet of Busto family has a
tail.
·Example 2: If a number is even  it is divisible by two.
- 120 is an even number  120 is divisible by two.
·Example 3: It rains today  the streets will be wet.
- Tomorrow will rain  the streets will be wet tomorrow.
2. Law of Syllogism: If P  Q is true and Q  R is true,
then P  R is true.
·Example 1: If an animal is a fish  it lives in water, if an animal
lives in water  it is wet all the time.
- A shark is a fish  a shark is wet all the time.
·Example 2: If an animal is a mammal  it has warm blood, if an
animal has warm blood  it can control it’s inner temperature.
- A dog is a mammal  it can control it’s inner temperature.
·Example 3: If a number is divisible by 12  it is divisible by 6, if a
number is divisible by 6  it is even.
- 36 is divisible by 12  36 is even.
Statements
Algebraic Proof
Reasons
1. 3x + 7 = 40
-7 -7
Given
3x / 11 = 33 / 11
Subtraction property of equality
X=3
Division property of equality
2. 5x – 8 = 42
+ 8 +8
Given
5x = 50
Addition property of equality
5x / 5 = 50 / 5
X = 10
Division property of equality
3. -3x + 4 = 34
-4 -4
Given
-3x / -3 = -30 / -3
Subtraction property of equality
X = - 10
Division property of equality
• You use the properties and work with the initial statement
until you reach the desired conclusion.
Properties of equality
Addition property
If a = b, then a + c = b + c
Subtraction property
If a = b, then a – c = b – c
Multiplication property
If a = b, then ac = bc
Division property
If a – b then c /= o, then a/c = b/c
Reflexive property
a=a
Symmetric property
If a = b, then b =a
Transitive property
If a = b and b = c, then a = c
Substitution property
If a = b, then b can be substituted for
a in any expression.
Properties of congruence
Reflexive property
--- --EF ~= EF
Symmetric property
If <1 ~= <2, then <1 ~= <2
Transitive property
If figure A ~= figure B and
figure B ~= figure C, then
figure A ~= figure C
Segments and Angles properties
You can work with the
measures of angles or
measures of sides using
the laws of algebra.
Property
Segments
Angles
Reflexive
PQ = PQ
M<1 = M<1
Symmetric
If AB = CD, then CD = AB If m<A = m<B, then
m<B = m<A
Transitive
If GH = JK and JK = LM,
then GH = LM
If m<1 = m<2 and m<2
= m<3, then m<1 = m<3
Property
Segments
Angles
Reflexive
PQ ~= PQ
M<1 ~= M<1
Symmetric
If AB ~= CD, then CD ~=
AB
If m<A ~= m<B, then
m<B ~= m<A
Transitive
If GH ~= JK and JK ~=
LM, then GH ~= LM
If m<1 ~= m<2 and m<2
~= m<3, then m<1 ~~=
m<3
P
Q
R
S
T
U
1. PQ = PQ
2. PQ = RS, then RS = PQ
3. If PQ = RS and RS = TU, then PQ =
RS
Proof process
1. Write the conjecture to be proven.
2. Draw a diagram to represent the hypothesis of the conjecture.
3. State the given information and mark it on the diagram.
4. State the conclusion of the conjecture in terms of the diagram.
5. Plan your argument and prove the conjecture.
Two proof- column
• You list your steps on
the proof in the left
column. You write the
matching reason for
each step in the right
column.
• They give you the given,
and what you need to
proof. Sometimes they
give you the plan.
Given: <1 and <2 are right angles
Prove: <1 ~= <2
Plan: Use the definition of a right angle to
write the measure of each angle. Then use
the Transitive Property and the definition
of congruent angles.
Statement
Reason
<1 and <2 are right angles
Given
M<1 = 90, m<2 = 90
Def or rt. Angles
M<1 = M<2
Trans prop. Of =
<1 ~= <2
Def. of ~= angles
Given: AB ~= XY
BC ~= YZ
Prove: AC ~= XZ
Statement
Reason
AB ~= XY, BC ~= YZ
Given
AB = XY, BC = YZ
Def. coongruent segments
AB + BC = XY + YZ
Addition porp of =
AB + BC = AC
XY + YZ = XZ
Segment Addition Postulate
AC = XZ
Subs. Prop of =
AC ~= XZ
Def. congruent segments
Given: PQ ~= RS
Prove: RS ~= PQ
Statement
Reason
PQ ~= RS
Given
PQ = RS
Def. of congruent segments
RS = PQ
Tsymmetric prop. =
RS ~= PQ
Def. of congruent segments
Linear pair postulate
• Linear pair: a pair of
adjacent angles whose
no common sides are
opposite rays.
EXAMPLES:
1.
2.
• LPP: If two angles form
a linear pair, then they
are supplementary
angles.
3.
Theorems
Theorem
Hypothesis
Conclusion
Congruent Supplements: If two angles
are supplementary to the same angle
(or two congruent angles), then the
two angles are congruent.
<1 and <2 are
supplementary.
<2 and <3 are
supplementary.
<1 ~= <3
Linear pair: if two angles form a linear
pair than they are supplementary.
<a and <b form a
linear pair.
<a and < b are
supplementary.
Right angle congruence: all right
angles are congruent,
<a and <b are
right angles.
<a ~= <b
Congruent complements: if two angles
are complementary to the same (or to
two congruent angles), then the two
angles are congruent.
<1 and <2 are
complementary.
<2 and <3 are
complementary.
<1 ~= <2
Vertical angles theorem
• Vertical angles are
congruent.
1.
2.
3.
Common Segment Theorem
• Give collinear points A,
B, C, and D arranged as
shown, if AB ˜= CD,
then AC ˜= BD.
EXAMPLES:
1.
• AB ˜= BD.
• AC ˜= BD.