Download Sect 2-1 Conditional Statements

Sect 2-1 Conditional Statements
Conditional statement- a statement that has two parts, a
hypothesis and a conclusion.
Ex. All sharks have a boneless skeleton.
If-then form of a conditional statement- the “if” part
contains the hypothesis and the “then” part contains the
conclusion.
Ex. If a fish is a shark, then it has a boneless skeleton.
Conditional statements may be true or false. A single
counterexample shows the statement is not always true.
Statement: If today is Monday, then tomorrow is Tuesday.
Converse of a conditional is formed by switching the
hypothesis and the conclusion.
Ex. Converse: If tomorrow is Tuesday, then today is
Monday.
Negation- writing the negative of a statement
Inverse of a conditional is formed by negating the
hypothesis and the conclusion of the statement.
Ex. Inverse: If today is not Monday, then tomorrow is not
Tuesday.
Contrapositive of a conditional is formed by negating the
hypothesis and conclusion of the converse of a conditional
statement.
Ex. Contrapositive : If tomorrow is not Tuesday, then today
is not Monday.
When two statements are both true or both false, they are
called logically equivalent statements.
A conditional and its contrapositive are logically equivalent
as are the converse and the inverse.
Postulate 5: Through any two points there exists exactly one
line.
Postulate 6: A line contains at least two points.
Postulate 7: If two lines intersect, then their intersection is
exactly one point.
Postulate 8: Through any three noncollinear points there
exists exactly one plane.
Postulate 9: A plane contains at least three noncollinear
points.
Postulate 10: If two points lie in a plane, then the line
containing them lies in the plane.
Postulate 11: If two planes intersect, then their intersection
is a line.
Sect 2-2
DEFINITIONS AND BICONDITIONAL STATEMENTS
What is a good definition?
A definition uses known words to describe a new word.
perpendicular lines- two lines that intersect to form a right angle.
All definitions in geometry can be interpreted “forward” and “backward.”
The conditional and the converse are both true in a good definition.
EX: Congruent angles have equal measures.
Conditional: If two angles are congruent, then their measures are equal.
Converse: If two angles have equal measures, then they are congruent.
Is the following a good definition?
A square has four right angles.
Biconditional statement: a statement that contains the phrase “if and
only if.”
Writing a biconditional is equivalent to writing a conditional and its
converse.
A biconditional statement can be either true of false. To be true, both the
conditional and the converse must be true.
All definitions can be written as true biconditional statements.
An angle is a right angle if and only if it measures 90.
EX: Rewrite the biconditional statement as a conditional and its converse.
Tom lives in Cincinnati if and only if he lives in Ohio.
True or false?
Geometry
Sec. 2.3 Deductive Reasoning
“p” is the hypothesis
“q” is the conclusion
 is “then”
pq is read “if p, then q”
The converse is qp
pq is read “p if and only if q” or “p iff q”
“~” is not
We use ~ to negate a hypothesis and/or a conclusion
The inverse would be ~p~q
The contrapositive would be ~q~p
Logical Argument – To use facts, definitions, and accepted properties
(otherwise known as deductive reasoning) in a logical order
Law of Detachment – If pq is true and p is true, then q must be true
Law of Syllogism – If pq is true and qr is true, then pr is true
Examples :
let p be “x is an even number” and let q be “x2 is even”
1. Write pq in words
2. Write qp in words
3. Write ~p~q in words
4. Write ~q~p in words
Is the argument valid
5. Jamal knows that if he misses the practice the day before the game,
then he will not be a starting player in the game. Jamal misses
practice on Tuesday so he concludes he will not start on Wednesday.
6. If two angles form a linear pair, then they are supplementary; angle 1
and angle 2 are a linear pair.
Geometry
Sec. 2.4 – Reasoning With Properties From Algebra
Algebraic 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 and c≠0, then a/c = b/c.
Reflexive Property – For any real number a, a=a.
For any segment AB = AB. For any angle m1 = m1.
Symmetric Property – For any real numbers a and b: if a=b, then b=a.
If AB = BC, then BC = AB. If m1 = m2, then m2 = m1.
Transitive Property – If a=b and b=c, then a=c.
If AB = BC and BC = CD, then AB = CD.
If m1 = m2 and m2 = m3, then m1 = m3.
Substitution Property – If a=b, then a can replace b in any equation.
Examples :
1. Solve and write the reasons for each step. 5x – 18 = 3x + 2
2. Solve and write the reasons for each step. 55z – 3(9z + 12) = -64
3. Solve for r and write each step. a = 220 – (10/7)r
4. Solve if a = 16 and write each step.
5. In the diagram AB = CD. Show AC = BD.
6. m1 + m2 = 66
m1 + m2 + m3 = 99
m3 = m1
m1 = m4
Find the m4 and write the reasons for each step.
Geometry
Sec. 2.5 Proving Statements About Segments
Definitions :
Theorem – A true statement that follows as a result of other true
statements. A theorem needs to be proved.
Two Column Proof – Has numbered steps on the left and
corresponding reasons on the right.
Paragraph Proof – A proof that can be written in paragraph form.
Thm. 2.1 Properties of Segment Congruence
Reflexive : For any segment AB, AB  AB
Symmetric : If AB  CD, then CD  AB.
Transitive : If AB  CD and CD  FG, then AB  FG.
Examples :
1. Given : PQ  XY
Prove : XY  PQ
2. Given : LK = 5, JK = 5, JK  JL
Prove : LK  JL
3. Given : Q is the midpoint of PR.
Prove : PQ = ½ PR and QR = ½ PR
Sec. 2.6 – Proving Statements About Angles
Theorem 2.2 – Properties of Angle Congruence
Reflexive – For any angle, A  A.
Symmetric – If A  B, then B  A.
Transitive – If A  B and B  C, then A  C.
Theorem 2.3 – Right Angle Congruence Theorem
All right angles are congruent.
Theorem 2.4 – Congruent Supplements Theorem
If two angles are supplementary to the same angle or
congruent angles, then they are congruent.
Theorem 2.5 – Congruent Complements Theorem
If angles are complementary to the same angle or
congruent angles, then they are congruent.
Postulate – Linear Pair Postulate
If two angles form a linear pair, then they are
supplementary.
Theorem 2.6 – Vertical Angle Theorem
Vertical angles are congruent.
Examples :
1.
2.
3.