GEOMETRYChapter 2 Notes

Section 2-1Conditional Statement: an if – then

statementEX: If it rains today, then the baseball

game will be canceled. Hypothesis: part after the ifConclusion: 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.

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

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.

Conditional: If x = 2, then |x| = 2.

Converse: If |x| = 2, then x = 2.

Symbolic:If p, then q. p qIf q, then p. q p

Section 2-2Biconditional: 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.

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.

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

Good definitionsUses clearly understood termsPrecise – avoid words like large, sort of,

someReversible – 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.

Section 2-3Deductive 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 <

90. Angle B is acute. Angle B’s measure must be < 90.

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.

Use the Law of Syllogism then the Law of Detachment to make a conclusion:1. If the circus is in town, then there are

tents at the fairgrounds. 2. If there are tents at the fairgrounds,

then Paul is working security.3. 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.

1. The Volga River is in Europe.

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

3. 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.

Section 2-4PROPERTIES 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 PropertyIf a = b and c ≠ 0, then a/c = b/c.

PROPERTIES OF EQUALITY

Reflexive Propertya = a

Symmetric PropertyIf a = b, then b = a.

Transitive PropertyIf a = b and b = c, then a = c.

Substitution PropertyIf a = b, then b can replace a in any

expression.

Distributive Propertya(b + c) = ab + ac

PROPERTIES OF CONGRUENCE

Reflexive Property or

Symmetric Property If , then .(same with

angles)

Transitive PropertyIf and , then .

€

AB ≅ AB

€

∠A ≅∠A

€

AB ≅ CD

€

CD ≅ AB

€

AB ≅ CD

€

CD ≅ EF

€

AB ≅ EF

JUSTIFYING YOUR STEPS

Must show EACH step (everything you do).

Must give a reason for why you are allowed to do what you did.PropertiesDefinitionsPostulates/Theorems