1. GEOMETRY
Chapter 2 Notes

2. Section 2-1 Conditional Statement: an if – then statement
EX: If it rains today, then the baseball game will be canceled.
Hypothesis: part after the if
Conclusion: part after the then
A rectangle has four right angles.
If a figure is a rectangle, then it has four right angles.
Subject of the sentence becomes the HYP.
Predicate of the sentence becomes the CONCL.

3. An integer that ends with 0 is divisible by 5.
If an integer ends in 0, then it is divisible by 5.
Truth value: whether the statement is true or false
Show a Counterexample:
If it is February, then there are only 28 days in the month.
It could be leap year!!!
If the name of a state contains the NEW, then the state borders an ocean.
New Mexico

4. Venn Diagrams
Converse of a Conditional: a statement that switches the hypothesis and the conclusion of a conditional.
Conditional: If two lines are not parallel and do not intersect, then they are skew.
Converse: If two lines are skew, then they are not parallel and do not intersect.

5. Symbolic: Conditional: If x = 2, then |x| = 2.
Converse: If |x| = 2, then x = 2.
Symbolic:
If p, then q. p  q
If q, then p. q  p

6. Section 2-2
Biconditional: a combination of a conditional and its converse when they are BOTH TRUE.
Uses the words if and only if (can shorten to iff)
Cond: If 2 angles have the same measure, then they are congruent.
Conv: If 2 angles are congruent, then they have the same measure.
Bicond: 2 angles have the same measure iff they are congruent.

7. Make a biconditional:
Cond: If 3 points are collinear, then they lie on the same line.
Conv: If 3 point lie on the same line, then they are collinear.
Bicond: 3 points are collinear iff they lie on the same line.

8. Bicond: A number is divisible by 3 iff the sum of its digits is a multiple of 3.
Cond: If a number is divisible by 3, then the sum of its digits is a multiple of 3.
Conv: If the sum of a numbers digits is a multiple of 3, then it is divisible by 3.
p iff q p <--> q

9. Good definitions Uses clearly understood terms
Precise – avoid words like large, sort of, some
Reversible – can be written as a biconditional
Show that the definition of perpendicular lines is reversible.
If 2 lines are perpendicular, then they intersect to form right angles.
If 2 lines intersect to form right angles, then they are perpendicular.
2 lines are perpendicular if and only if they intersect to form right angles.

10. Section 2-3
Deductive Reasoning: the process of reasoning logically from given facts to a conclusion
Law of Detachment:
If a conditional is true, then when given an example of its hypothesis, the conclusion must be true.
If it’s a Saturday, then Mike is at work. It’s Saturday. Mike must be at work.
If an angle is acute, then its measure is < Angle B is acute. Angle B’s measure must be < 90.

11. If p  q and q  r are true, then p  r must be true.
Law of Syllogism:
If p  q and q  r are true, then p  r must be true.
Cut out the part that is the same, and squeeze together what’s left!
If a number ends in 0, then it’s divisible by 10.
If a number is divisible by 10, then it’s divisible by 5.
If a number ends in 0, then it’s divisible by 5.

12. Use the Law of Syllogism then the Law of Detachment to make a conclusion:
If the circus is in town, then there are tents at the fairgrounds.
If there are tents at the fairgrounds, then Paul is working security.
The circus is in town.
PROCESS: Put the 2 conditional statements together:
If the circus is in town, then Paul is working security.
Then make a conclusion using #3.
CONLUSION: Paul is working security.

13. The Volga River is in Europe.
If a river is less than 2,300 miles long, then it is not one of the world’s ten longest rivers.
If a river is in Europe, then it is less than 2,300 miles long.
If a river is in Europe, then it is not one of the world’s ten longest rivers.
The Volga River is less than 2,300 miles long.
The Volga River is not one of the world’s ten longest rivers.

14. PROPERTIES OF EQUALITY
Section 2-4
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.

15. PROPERTIES OF EQUALITY
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 replace a in any expression.

16. PROPERTIES OF CONGRUENCE
Distributive Property
a(b + c) = ab + ac
PROPERTIES OF CONGRUENCE
Reflexive Property or
Symmetric Property
If , then (same with angles)
Transitive Property
If and , then

17. Must show EACH step (everything you do).
JUSTIFYING YOUR STEPS
Must show EACH step (everything you do).
Must give a reason for why you are allowed to do what you did.
Properties
Definitions
Postulates/Theorems