**Chapter Two: Geometric Reasoning**

**Section One: Using Inductive Reasoning to Make Conjectures**

Inductive reasoning: process of reasoning that a rule or statement is true because specific cases are true.

Conjecture: A statement you believe to be true based on inductive reasoning. Example: The product of an even number and an odd number is ___________. 2x3=6 EVEN NUMBER

Counterexample: can be a drawing statement, or a number. Example: For any three points in a plane, there are three different lines that contain two of the points.

If three points are collinear then the conjecture is false.

Conjecture: A statement you believe to be true based on inductive reasoning. Example: The product of an even number and an odd number is ___________. 2x3=6 EVEN NUMBER

Counterexample: can be a drawing statement, or a number. Example: For any three points in a plane, there are three different lines that contain two of the points.

If three points are collinear then the conjecture is false.

**Section Two: Conditional Statements**

Conditional statement: A statement that can be written in the form “if p, then q.” Example: If today is Wednesday, then tomorrow is Thursday.

Hypothesis: the part p of an conditional statement following the word if. Example: Today is Wednesday.

Conclusion: the part q of a conditional statement following the word then. Example: Tomorrow is Thursday.

Truth value: A conditional statement has a truth value of either true or false. It is false only when the hypothesis is true and the conclusion is false. Example: If I don’t get paid, I haven’t broken my promise.

Converse: a statement formed by exchanging the hypothesis and conclusion. Example: If tomorrow is Thursday, then today is Wednesday.

Inverse: a statement formed by negating the hypothesis and the conclusion. Example: If today is not Wednesday then tomorrow is not Thursday.

Contra positive: the statement formed by both exchanging and negating the hypothesis and conclusion. Example: If tomorrow is not Thursday then today is not Wednesday.

Logically equivalent statements: when conditional statements have the same truth value.

Hypothesis: the part p of an conditional statement following the word if. Example: Today is Wednesday.

Conclusion: the part q of a conditional statement following the word then. Example: Tomorrow is Thursday.

Truth value: A conditional statement has a truth value of either true or false. It is false only when the hypothesis is true and the conclusion is false. Example: If I don’t get paid, I haven’t broken my promise.

Converse: a statement formed by exchanging the hypothesis and conclusion. Example: If tomorrow is Thursday, then today is Wednesday.

Inverse: a statement formed by negating the hypothesis and the conclusion. Example: If today is not Wednesday then tomorrow is not Thursday.

Contra positive: the statement formed by both exchanging and negating the hypothesis and conclusion. Example: If tomorrow is not Thursday then today is not Wednesday.

Logically equivalent statements: when conditional statements have the same truth value.

**Section Three: Using Deductive Reasoning to Verify Conjectures**

Deductive reasoning: the process of using logic to draw conclusions from given facts,definitions, and properties.

Law of Detachment: If p to q is a true statement and p is true, then q is true. example: If you are tardy 3 times you must go to detention. Shea is in detention. Conjecture: Shea is tardy three times. “Shea is in detention” matches the conclusion of a true conditional though it doesn’t mean the hypothesis is true. Shea could also be in detention for something else. The conjecture is invalid.

Law of Syllogism: If p to q and q to r are true statements, then p to r is a true statement. example: If a number is divisible by 4, then it is divisible by 2. If a number is even then it is divisible by 2. Conjecture: If a number is divisible by 4, then it is even. This is not valid because the conclusion of both conditionals can not be used to draw a conclusion. The logic here used is not valid.

Applying the law of deductive reasoning: Drawing conclusion

Example: If a team wins 10 games, then they play to the finals. If a team plays in the finals, then they travel to Boston. The Ravens won ten games. CONCLUSION: The Ravens will travel to Boston.

Law of Detachment: If p to q is a true statement and p is true, then q is true. example: If you are tardy 3 times you must go to detention. Shea is in detention. Conjecture: Shea is tardy three times. “Shea is in detention” matches the conclusion of a true conditional though it doesn’t mean the hypothesis is true. Shea could also be in detention for something else. The conjecture is invalid.

Law of Syllogism: If p to q and q to r are true statements, then p to r is a true statement. example: If a number is divisible by 4, then it is divisible by 2. If a number is even then it is divisible by 2. Conjecture: If a number is divisible by 4, then it is even. This is not valid because the conclusion of both conditionals can not be used to draw a conclusion. The logic here used is not valid.

Applying the law of deductive reasoning: Drawing conclusion

Example: If a team wins 10 games, then they play to the finals. If a team plays in the finals, then they travel to Boston. The Ravens won ten games. CONCLUSION: The Ravens will travel to Boston.

**Section Four: Biconditional Statements And Definition**

A biconditional statement: a statement that can be written in the form “p if and only if q.”

Definition: a statement that describes a mathematical object and can be written a a true biconditional.

Polygon: defined as a closed plane figure formed by three or more line segments.

Triangle: a three-sided polygon.

Quadrilateral: a four-sided polygon.

Definition: a statement that describes a mathematical object and can be written a a true biconditional.

Polygon: defined as a closed plane figure formed by three or more line segments.

Triangle: a three-sided polygon.

Quadrilateral: a four-sided polygon.

**Section Five: Algebraic Proof**

**Properties of Equality:**

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 and c does not equal 0 then a/c = b/c

Reflexive Property of Equality: a=a

Symmetric Property of Equality: If a=b then b=a

Transitive Property of Equality: a=b and b=c then a=c

Substitution Property of Equality: If a=b, then b can be substituted for a in any expression.

**Properties of Congruence:**

Reflexive Property of Congruence: Figure A is congruent to Figure A

Symmetric Property of Congruence: Figure A is congruent to Figure B then Figure B is congruent to Figure A.

Transitive Property of Congruence: Figure A is congruent to Figure B and Figure B is congruent to Figure C then Figure A is congruent to Figure C.

**Section Six: Geometric Proof**

Theorem: a statement that can be proved.

2-6-1: Linear Pair Theorem: If two angles form a linear pair, then they are supplementary.

Hypothesis- Angle A and Angle B form a linear pair. Conclusion- Angle A and Angle B are supplementary.

2-6-2: Congruent Supplements Theorem: If two angles are supplementary to the same angle then the two angles are congruent.

Hypothesis: Angle 1 and Angle 2 are supplementary and Angle 2 and Angle 3 are supplementary.

Conclusion: Angle 1 is congruent to Angle 3.

Two column proof: Steps on left column and matching reason on right column.

2-6-3: Right Angle Congruence Theorem: All right angles are congruent.

Hypothesis: Angle A and angle B are right angles.

Conclusion: Angle A is congruent to Angle B.

2-6-4: Congruent Complements Theorem:If two angles are complementary to the same angle then the two angles are congruent.

Hypothesis: Angle 1 and Angle 2 are complementary and Angle 2 and Angle 3 are complementary.

Conclusion: Angle 1 is congruent to Angle 3.

2-6-1: Linear Pair Theorem: If two angles form a linear pair, then they are supplementary.

Hypothesis- Angle A and Angle B form a linear pair. Conclusion- Angle A and Angle B are supplementary.

2-6-2: Congruent Supplements Theorem: If two angles are supplementary to the same angle then the two angles are congruent.

Hypothesis: Angle 1 and Angle 2 are supplementary and Angle 2 and Angle 3 are supplementary.

Conclusion: Angle 1 is congruent to Angle 3.

Two column proof: Steps on left column and matching reason on right column.

2-6-3: Right Angle Congruence Theorem: All right angles are congruent.

Hypothesis: Angle A and angle B are right angles.

Conclusion: Angle A is congruent to Angle B.

2-6-4: Congruent Complements Theorem:If two angles are complementary to the same angle then the two angles are congruent.

Hypothesis: Angle 1 and Angle 2 are complementary and Angle 2 and Angle 3 are complementary.

Conclusion: Angle 1 is congruent to Angle 3.