Geometric Proofs

Front 1

Inductive Reasoning

Back 1

The process reasoning that a rule or statement is true because specific cases are true.

Front 2

Conjecture

Back 2

A statement you believe to be true based on inductive reasoning.

Front 3

Counterexample

Back 3

An example that proves the conjecture false.

Front 4

Example of Conjecture

Back 4

The product of a number squared is always an even number.

Front 5

Example of Counterexample

Back 5

The product of a number squared is always an even number.

A counterexample would be 3.
(3 x 3 = 9)

Front 6

Conditional Statement

Back 6

A statement that can be written in the form "if p then q"

Front 7

Hypothesis (geometry definition)

Back 7

The part "p" of a conditional statement, following the word "if"

p (hypothesis) -> q

Front 8

Conclusion (geometry definition)

Back 8

The part "q" of a conditional statement, following the word "then"

p -> q (conclusion)

Front 9

Truth Value

Back 9

Tells whether a conditional statement is true (T) or false (F). It is false only when the hypothesis is true and conclusion is false. The truth value is true if the conclusion is true.

Front 10

Negation

Back 10

The negation of statement "p" is written as "~p". The negation of a true statement is false and the negation of a false statement is true.

Front 11

Converse of Conditional Statement

Back 11

The statement formed by exchanging the hypothesis and conclusion. In other words, you flip them.

q -> p

Front 12

Inverse of Conditional Statement

Back 12

The statement formed by negating the hypothesis and the conclusion. You do this by proving the hypothesis and conclusion wrong.

~p -> ~q

Front 13

Contrapositive of Conditional Statement

Back 13

The statement formed by both exchanging and negating the hypothesis and conclusion.

~q -> ~p

Front 14

Logically Equivalent Statements

Back 14

Related Conditional Statements that have the same truth value. A conditional and its contrapositive are equivalent, and so are the converse and inverse.

Ex: If the converse (q -> p) and the inverse (~p -> ~q) are both false, they are logically equivalent.

Front 15

Example of Negation

Back 15

(hypothesis) If an animal has four paws, (conclusion) it is a dog.

A cat also has four paws, so this statement is false.

Front 16

Deductive Reasoning

Back 16

The process of using logic to draw conclusions from given facts, defintions, and properties.

Front 17

Law of Detachment

Back 17

The Law of Detachment is one valid form of deductive reasoning.

If p -> q is a true statement, then p is true and q is true.

Front 18

Law of Syllogism

Back 18

Another form of valid deductive reasoning. This method allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other.

If p -> q and q -> r are true statements, then p -> r is a true statement.

Front 19

Example of using the Law of Detachment in Conditional Statements

Back 19

Given: If you read ten books, then you get a reward. Tom gets a reward.
Conjecture: Tom gets a reward.

The conjecture matches the conclusion, but that does not mean the hypothesis is true. Tom could have gotten a reward for something else.

Front 20

Biconditional Statement

Back 20

A statement that can be written in the form "p if and only if q." This means "if p, then q" and if "if q, then p."

p <--> q means p -> q and q -> p

Front 21

Definition (geometry definition)

Back 21

A statement that describes a mathematical object and can be written as a true biconditional.

Front 22

Example of Biconditional

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(hypothesis [p]) A polygon is a square if and only (conclusion [q]) if it has four congruent sides.

Front 23

Definition

Back 23

A statement that describes a mathematical object and can be written as a true biconditional. A good definition can be used forwards and backwards. For example, if a figure is a quadrilateral, then it is a four sided polygon. If a figure is a four sided polygon, it is a quadrilateral.

Front 24

Proof (geometry definition)

Back 24

An arguement that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.

Front 25

Addition Property of Equality

Back 25

If a = b, then a + c = b + c

EX: a = 5
b = 5 (a) 5 + 10 = (b) 5 + 10
c = 10

Front 26

Subtraction Property of Equality

Back 26

If a = b, then a - c = b - c

EX: a = 5
b = 5 (a) 5 - 2 = (b) 5 - 2
c = 2

Front 27

Multiplication Property of Equality

Back 27

If a = b, then ac = bc

EX: a = 5
b = 5 (a) 5 x 10 = (b) 5 x 10
c = 10

Front 28

Division Property of Equality

Back 28

If a = b, and c =/= 0, then a/c = b/c

EX: a = 20
b = 20 (a) 20/10 = (b) 20/10
c = 10

Front 29

Reflexive Property of Equality

Back 29

a = a
(Technically, it's the same point/angle)

EX: 5 = 5

Front 30

Symmetric Property of Equality

Back 30

If a = b, then b = a

Front 31

Transitive Property of Equality

Back 31

If a = b and b = c, then a = c

Front 32

Substitution Property of Equality

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If a = b, then b can be substituted for a in any expression

EX: If b = 5x, and a = 2b <-------
(b = 5x), then you can substitute "b" into "a". So "a" would equal 2(5x) or 10x.

Front 33

Properties of Equality

Back 33

Algebraic properties that are used to solve algebraic proof.

Front 34

Symmetric Property of Congruence

Back 34

If figure A is congruent to figure B, then figure B is congruent to figure A.

Front 35

Reflexive Property of Congruence

Back 35

Figure A is congruent to figure A

Front 36

Transitive Property of Congruence

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If figure A is congruent to figure B and figure B is congruent to figure C, then figure A is congruent to figure C.

Front 37

Theorem

Back 37

Any statement that you can prove.

Front 38

Linear Pair Theorem

Back 38

If two angles form a linear pair, then they are supplementary.

Front 39

Congruent Supplements Theorem

Back 39

If two angles are supplementary to the same angle, then the two angles are congruent.

Front 40

Two-Column Proof

Back 40

A way of solving a proof in which you put the steps of the proof in the left column and the matching reason for each step in the right column.

Front 41

Right Angle Congruence Theorem

Back 41

All right angles are coungruent.

Front 42

Congruent Complements Theorem

Back 42

If two angles are complementary to the same angle, then the two angles are congruent.

Front 43

Flowchart Proof

Back 43

A way of solving a proof that uses boxes and arrows to show the structure of the proof.

Front 44

Vertical Angles Theorem

Back 44

Vertical angles are congruent.

Front 45

Theorem 2-7-3

Back 45

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