Geometry Definitions, Postulates, and Theorems

Geometry Definitions, Postulates, and Theorems

Chapter 2: Reasoning and Proof

Section 2.1: Use Inductive Reasoning

Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.

Conjecture -

Ex. What’s next? · Visual Pattern a)

· Number Pattern a)

Inductive Reasoning -

Ex. The product of an odd number and an even number is .

Ex. The sum of an odd number and an even number is .

Counterexample –

Ex. The difference between two positive numbers is always positive. Find a counterexample.

(there are many correct answers)

Section 2.2: Analyze Conditional Statements

Standards: 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.

Conditional Statement –

Hypothesis –

Conclusion –

Ex. Rewrite the conditional statements in if-then form:

a) It is time for dinner if it is 6pm.

b) All monkeys have tails.

*Conditional statements can be true or false. Decide whether the statement is true or false. If

false, provide a counterexample.

Ex. If a number is odd, then it is divisible by 3.

Converse –

Ex. Statement: If you see lightening, then you hear thunder.

Converse:

Negation –

Ex. a) , b) is obtuse,

Inverse –

Ex. Statement: If an animal is a dog, then it has four legs.

Inverse:

Contrapositive –

Ex. Statement: If an animal is a fish, then it can swim.

Contrapositive:

Equivalent Statements -

Ex. a) Statement: If , then is acute.

Contrapositive:

b) Inverse:

Converse:

***Perpendicular Lines –

Biconditional Statement –

Section 2.3: Apply Deductive Reasoning

Deductive Reasoning –

Inductive Reasoning –

*Two types of Deductive Reasoning:

Law of Detachment –

Ex.

Law of Syllogism –

Ex.

Ex. Decide whether deductive or inductive reasoning is being used in the following argument.

Everyday Mrs. Maya reads off the homework answers, then goes over the wrong ones on the board. Today she has read off the homework answers, so you can conclude that she will go over the wrong answers on the board.

Why did you decide so?

Ex. Law of Syllogism

If you are a student, then you have lots of homework.

If you have lots of homework, then you have little social life.

Section 2.3 Extension

Symbolic Notation: p represents the hypotheses

q represents the conclusion

conditional statement (if-then):

converse

negation of p

inverse

contrapositive

biconditional (means and )

Ex. Use p and q to write the symbolic statement in words.

p: A number is divisible by 3.

q: A number is divisible by 6.

Section 2.4: Use Postulates and Diagrams

Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

Postulate 5 –

Postulate 6 –

Postulate 7 –

Postulate 8 –

Postulate 9 –

Postulate 10 –

Postulate 11 –

Line Perpendicular to a Plane –

Ex. State the postulate that verifies the truth of the statement: V and T lie on line .

Section 2.5: Reason Using Properties from Algebra

Addition Property of Equality – IF a = b,

THEN a + c = b + c.

Subtraction Property of Equality – IF a = b,

THEN a – c = b – c.

Multiplication Property of Equality – IF a = b,

THEN ac = bc.

Division Property of Equality – IF a = b,

THEN a ¸ c = b ¸ c.

***Substitution Property of Equality – IF a = b,

THEN a can be substituted for b anytime.

Distributive Property – OR

Properties of Equality for Real Numbers, Segments and for Angles

Real Numbers Segments Angles

***Reflexive IF a is a real # IF Segment AB, IF Angle A,

THEN a = a THEN AB = AB. THEN .

***Symmetric IF a = b IF , IF ,

THEN b = a THEN . THEN .

***Transitive IF a = b IF , IF ,

AND b = c, AND , AND ,

THEN a = c. THEN . THEN .

(over)

Ex. Algebraic Proof – Solve the equation and state the reason for each step.

Ex. Geometric Proof – Given:

Prove:

Ex. Use the property to complete the statement.

Transitive Property: If and , then

Section 2.6 – Prove Statements about Segments & Angles

Standards: 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2.0 Students write geometric proofs, including proofs by contradiction.

Proof –

Two-Column Proof –

Statements Reasons

1) 1)

2) 2)

etc) etc)

Theorem –

***Theorem 2.1 Congruence of Segments is Reflexive

GIVEN: Segment AB,

THEN AB @ AB.

***Theorem 2.1 Congruence of Segments is Symmetric

IF ,

THEN .

***Theorem 2.1 Congruence of Segments is Transitive

IF ,

AND ,

THEN .

Ex. Given: P Q

Prove: X Y

(over)

Ex. Given: Q is the midpoint of . P Q R

Prove: PQ =

Ex. Given: EF = GH

Prove: EG = FH E F G H

Ex. Given:

Prove:

Ex. In the diagram, if , find BC.

***Theorem 2.2 Congruence of Angles is Reflexive

GIVEN: Angle A,

THEN.

***Theorem 2.2 Congruence of Angles is Symmetric

IF ,

THEN .

***Theorem 2.2 Congruence of Angles is Transitive

IF ,

AND ,

THEN .

Ex. Proof of Transitive Property of Congruence

Given:

Prove: A B C

Statements Reasons

1) 1) given

2) 2)

3) 3)

4) 4)

Ex. Given:

Prove:

Ex. Given:

Prove:

Section 2.7 – Prove Angle Pair Relationships

***Theorem 2.3 Right Angle Congruence Theorem

Ex. Given: are right angles,

Prove:

***Theorem 2.4 Congruent Supplements Theorem

***Theorem 2.5 Congruent Complements Theorem

Ex. Given:

Prove:

***Linear Pair Postulate –

***Theorem 2.6 Vertical Angles Theorem

Vertical angles are congruent.

Ex. Given: are a linear pair, are a linear pair

Prove:

Ex. See diagram

c) If

d) If

f) If