Download Geometry A Unit 2 Day 2 Notes 2.2: Reasons in Mathematics I

Geometry A Unit 2 Day 2 Notes
2.2: Reasons in Mathematics
I. Arguing your point is an important skill. It is made easier when there is a shared vocabulary.
Your arguments are stronger when you avoid assumptions and opinion. Stick to definitions
and undeniable logic.
A. Definitions we’ve used in this course.
1. Congruent
2. Midpoint
3. Bisect
4. Bisector
What was the common thread running through the definitions of the terms above?
________________________________________________________________.
5. Complementary
6. Supplementary
B. A new definition
1. Perpendicular - _________________________________________________
A
a. The symbol for perpendicular is _________.
H
Ex. 1: Name the two pairs of perpendicular segments
in the figure to the right.
_______  _________,
B
E
F
___________________
b. Although the definition doesn’t say it, we know
a lot of angles in a diagram if it has perpendicular lines.
C
D
Ex. 2: Find each of the following angles in the diagram above.
 EFD = _____________
 DFB = ________________
Assume for the next three problems that  HFA = 25
 AFB = ___________
 AFE = _______________
 CFD = _____________
II. To make good arguments, we need to use definitions and postulates to understand conditional
statements and logic.
1. Old Ideas
A. Conditional – Any IF...THEN
B. Postulate – A statement that is accepted as true without proof.
Ex: Segment Addition Postulate
IF B is between A and C, AB + BC = AC
Ex: If two planes intersect, their intersection is a line.
C. Definitions – Can be read backwards and forward.
Ex. If two lines are perpendicular, then they intersect to form a right angle AND
If two lines intersect to form a right angle, then they are perpendicular.
2. New Vocabulary/Ideas
A. Biconditional – A statement containing the phrase ________________________.
1) A biconditional statement quickly shows that a statement and its
________________________ are both true.
Ex: Write the following as a conditional and its converse.
The ceiling fan runs if and only if the light switch is on.
Conditional: ____________________________________________________________
____________________________________________________________
Converse: ______________________________________________________________
______________________________________________________________
Ex: Decide if the biconditional statements are true. If they are not, give a
counterexample.
1) An angle is obtuse if and only if it measures more than 90o.
2) Two angles form a vertical pair of angles if and only if they have the same measure.
III. Symbolic Notation and Deduction
A. Earlier, we used p and q to represent parts of a conditional. We are going to add to the
notation now.
1. The symbolic notation for If p, then q is ___________________
2. The symbolic notation for p if and only if q is _________________.
Ex: Let p be “the value of x is -5” and q be “the absolute value of x is 5.”
a) Write p q in words. ________________________________________________
___________________________________________________________________
b) Write q  p in words ________________________________________________
___________________________________________________________________.
c) Decide whether the biconditional p  q is true. _________________________
B. Deductive Thinking - using facts, definitions, and accepted properties in a logical order to
write a logical argument.
(Compared to Inductive Thinking – using patterns to form conjectures)
1. Two basic pieces of logic
a) Law of Detachment – If we know one If …then is true, and ______
_____________________________________________________
_______________________________________________________
Notation: Given: __________________________, we conclude __________.
b) Law of Syllogism – If we know two If …then’s are true, and the then in one
is the if in the other, then ________________________
____________________________________________.
Notation: Given: ____________________________________, we conclude ________.
2. Examples
a) State whether the argument is valid.
Janice knows that if she is not in school the day before a game, she will not start the next
day. She was not in school on Tuesday, so she concludes she will not be able to start on
Wednesday.
If two angles form a linear pair, then they are supplementary.  A and  B are
supplementary. We can conclude that  A and  B are a linear pair.
b) Make a conclusion using the Law of Syllogism. (You won’t need all of the statements)
If a bird is the fastest bird on land, then it is the largest of all birds.
If a bird is the largest of all birds, then it is flightless.
If a bird is an ostrich, then it is flightless.
HW: p. 82 #13-19 all, 22, 27-29
p. 91 #9, 12, 21-24, 30
If you do not write down all of the questions on your homework, you MUST bring your books to class tomorrow
for homework review