Download p q : If two angles are vertical, then they are congruent.

Schema:
Recursive Rule
Explicit Rule
1. Generate a sequence using the description, then
write a rule to predict the n-th term:
“ The first term in the sequence is 4 and each
term is three more than twice the previous term.”
1.3 The Axiomatic System of
Geometry
Solve Problems
(Formally)
Theorems
Definitions & Properties
Postulates & Undefined Terms
Read & do #1 - #3
(then check answers in your group)
Properties of Equality
Distributive Property of
Multiplication over addition (and
subtraction?)
Read & do #4
Check the answer in your group
Two Column Proof
Try These A a)
1)Read Example 2 & do Try These B
2) then do #5
3) Stop before #6
Example 2 & Try These B:
Schema:
Provide a 2 column proof for solving
this equation
Formative B
only if = then
implies = then
Symbolic Logic
pq
Example:
pq:
is used to represent
if p, then q
or
p implies q
p: a number is prime
q: a number has exactly two divisors
If a number is prime, then it has exactly two divisors.
Symbolic Logic - continued
~
is used to represent the word
Example 1:
p: the angle is obtuse
~p:
The angle is not obtuse
“not”
Note:
~p means that the angle could be acute, right, or
straight.
Example 2:
p: I am not happy
~p:
I am happy
~p took the “not” out- it would have been a double negative (not not)
Observations:
• Conditional statements can be either true or
false.
• To show that a conditional statement is true,
you must provide (write) a proof.
• To show that a conditional statement is false,
you must describe a single example that shows
the statement is not always true.
• The example that shows a statement is false is
a COUNTEREXAMPLE.
DO #6
Forms of Conditional Statements
Converse: Switch the hypothesis and conclusion (q  p)
pq
If two angles are vertical, then they are congruent.
qp
If two angles are congruent, then they are vertical.
27
Forms of Conditional Statements
Inverse: State the opposite of both the hypothesis and conclusion.
(~p~q)
pq : If two angles are vertical, then they are congruent.
~p~q: If two angles are not vertical, then they are not
congruent.
Forms of Conditional Statements
Contrapositive: Switch the hypothesis and conclusion and
state their opposites. (~q~p)
pq : If two angles are vertical, then they are congruent.
~q~p: If two angles are not congruent, then they are not
vertical.
Do #7, then check the answers in your group
Forms of Conditional Statements
• Contrapositives are logically equivalent to the
original conditional statement.
• If pq is true, then qp is true.
• If pq is false, then qp is false.
Biconditional
• When a conditional statement and its converse are both true,
the two statements may be combined.
• Use the phrase if and only if (sometimes abbreviated: iff)
Statement: If an angle is right then it has a measure of 90.
Converse: If an angle measures 90, then it is a right angle.
Biconditional: An angle is right if and only if it measures 90.
Biconditional
• All definitions can be written as biconditional
statements.
• 𝑝
𝑞
𝑎𝑛𝑑
𝑞
𝑡𝑟𝑢𝑒
𝑝
𝑝
𝑞
Do #8,9,10 &Try These C
HW:
• Try These C
• P30, Check your understanding #1-13