Chapter 2 Geometry Notes

2.1/2.2 Patterns and Inductive Reasoning and Conditional Statements

Inductive reasoning: looking at numbers and determining the next one

Conjecture: sometimes thought of as an educated guess; you must gather as much information as you can to make a conjecture

Examples: Find the next two terms in each sequence.

1.) 12, 17, 22, 27, 32, …

2.) 5000, 1000, 200, 40, …

3.) 1, 12, 123, 1234, …

4.) 1, 4, 9, 16, 25, …

Draw the next figure in the sequence.

5.)

6.) For a science experiment, you measure the height of a plant every two days. Using inductive reasoning and the data table at the right, predict the height of the plant on day 10 of that experiment.

Counterexamples: ______________________________________________________________

Example: I tell you that every odd number is divisible by 3. Prove me wrong with a counterexample.

Find one counterexample to show that each conjecture is false.

7.) The result of a number multiplied by a positive integer is always larger than the original number.

8.) February has exactly 28 days every year.

Conditional: ___________________________________________________________________

Parts to Conditional Statements

1.) _______________________________________________________________________ 2.) ________________________________________________________________________

Symbolic Form: fill in the blanks.

Conditional statements can be written using the letters __________ and __________.

These letters represent the ____________________ and the ____________________.

The conditional p q in symbols reads “If p, then q”

The “p” represents the hypothesis and the “q” represents the conclusion.

Examples: For 1-3, identify the hypothesis and conclusion of each conditional by underlining the hypothesis and circling the conclusion.

1.) If you can predict the future, then you can control the future.

2.) If Dan is nearsighted, then Dan needs glasses.

3.) If lines k and m are skew, then lines k and m are not perpendicular.

Negation: _____________________________________________________________________

Symbolically: negating the statement p would be ______________________________________

Related Statements to Conditionals: Each conditional has 3 related statements.

1.) Converse: _______________________________________________________________ Symbols:

2.) Inverse: ________________________________________________________________ Symbols:

3.) Contrapositive:___________________________________________________________ Symbols:

Equivalent Statements: __________________________________________________________

Conditionals and Contrapositives are equivalent.

Converses and Inverses are equivalent.

Examples: If the given statement is not in if-then form, rewrite it. Write the converse, inverse, and contrapositive of each conditional statement. Determine the truth value of each statement.

4.) If 3x – 7 = 20, then x = 9.

5.) Baseball players are athletes.

2.3 Biconditional Statements

Biconditionals: _________________________________________________________________

Writing Biconditionals

1.) Make certain both the conditional and converse are true statements. 2.) Identify the hypothesis and conclusion. 3.) Write a new conditional with the hypothesis, the phrase “if and only if” and the

conclusion

For example: I go to the game if and only if my husband pays for popcorn.

Examples: For each of the statements, write the conditional form and then the converse of the conditional. If the converse is true, combine the statements as a biconditional.

1.) If the sum of two angles is 180, then the two angles are supplementary.

2.) Rectangles have four sides.

3.) Three points on the same line are collinear.

Examples: Write the two statements that form each biconditional. Tell whether each statement is true or false.

4.) Lines m and n are skew if and only if lines m and n do not intersect.

5.) A person can be president of the United States if and only if the person is a citizen of the United States.

Good Definitions: If a definition is not good, we can use ______________ to prove that it is not.

Reversible Definitions: Definitions whose conditional and converse are true statements. In other words, reversible definitions can be written as __________________________________.

Example:

6.) Is the definition of a quadrilateral reversible? If yes, write it as a biconditional. Definition: A quadrilateral is a polygon with four sides.

Examples: Is each statement a good definition? If not, find a counterexample.

7.) Segments with the same length are congruent.

8.) A fish is an animal that swims.

2.5 Reasoning in Algebra and Geometry

Postulates:_____________________________________________________________________

Rules or properties from algebra can be thought of as postulates also.

Geometry is reasoning, know properties from algebra helps in reasoning steps.

Property of Equality Definition Example

Addition

Subtraction

Multiplication

Distributive

Division

Reflexive

Symmetric

Transitive

Substitution

Example:

1.) Solve for x. Show your work. Justify each step.

Given: Ray PF bisects <EPG.

Properties of Congruence: Some properties work as equality and congruence properties.

Property of Congruence Definition Example

Reflexive

Symmetric

Transitive

Examples: Name the property of equality of congruence that justifies going from the first statement to the second statement.

2.) <M <N <N <M

3.) 3x = 24 X = 8

4.)

Proofs: ________________________________________________________________________

Two Column Proof: ______________________________________________________________

Organize your thoughts and steps into a progression.

Example 5: Write a two-column proof.

Given: m<1 = m<3

Prove: m<AEC = m<DEB

Statements Reasons

Example 6: Write a two column proof.

Given: CDAB

Prove: BDAC

Statements Reasons

2.6 Proving Angles Congruent

Theorem: _____________________________________________________________________

Vertical Angles Theorem: vertical angles are always congruent

Diagram:

Find x in the following.

1.)

2.)

3.)

ALL PROPERTIES DISCUSSED BEFORE APPLY TO ANGLES AS WELL. BRUSH UP ON THEM!!!!!!!!!!!

Congruent Supplements Theorem: If two angles are supplements of the same angle or congruent angles, then the two angles are congruent.

Diagram:

Congruent Complements Theorem: If two angles are complements of the same angle or congruent angles, then the two angles are congruent.

Diagram:

Theorem: All right angles are congruent.

Diagram:

Theorem: If two angles are congruent and supplementary, then each is a right angle.

Diagram:

Using a Paragraph Proof

Instead of two-columns, the proof is laid out in sentences.

Proof of the Congruent Supplements Theorem

Given: <1 and <3 are supplementary.

<2 and <3 are supplementary.

Prove: 21

<1 and <3 are supplementary because ______________. Therefore m<1 +m<3 = 180 by the ______________. <2 and <3 are also supplementary because of the given. So, m<2 + m<3 = 180 by the ______________. So, m<1 + m<3 = m<2 + m<3 by the ______________. So, m<1 =

m<2 by the _____________. Angles with the same measure are _____________ and 21 .