# Proofs

## Big Picture

Mathematical reasoning and proofs are a fundamental part of geometry. Several tools used in writing proofs will be covered, such as reasoning (inductive/deductive), conditional statements (converse/inverse/contrapositive), and congruence properties. The purpose of a proof is to prove that a mathematical statement is true.

## Key Terms

Proof: A logical argument that uses logic, definitions, properties, and previously proven statements to show a statement is true.

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

Postulate: Basic rule that is assumed to be true. Also known as an axiom.

Theorem: Rule that is proven using postulates, definitions, and other proven theorems.

Congruent: When two geometric figures have the same shape and size.

## Algebraic Proofs

Solving an algebraic equation is like doing an algebraic proof. Algebraic proofs use algebraic properties, such as the properties of equality and the distributive property.

Basic Algebraic Properties

### Property

Reflexive Property of Equality

a = a

Symmetric Property of Equality

a = b and b = a

Transitive Property of Equality

a = b and b = c, than a = c

Substitution Property of Equality

If a = b, then b can be used in place of a and vice versa

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

Distributive Property

a(b + c) = ab + ac

Although we may not write out the logical justification for each step in our work, there is an algebraic property that justifies each step.

Example:

- 3 = n + 1

Given

-3 -1 = n +1 -1

Subtraction Property of Eguality

-4 = n

Simplify

n = -4

Symmetric Property of  Equality.

Since segment lengths and angle measures are real numbers, the following properties of equality are true for segment lengths and angle measures:

• Reflexive Property of Equality
• Symmetric Property of Equality
• Transitive Property of Equality

## Geometric Proofs

A proof is a logical argument that shows a statement is true.

Definitions, postulates, properties, and theorems can be used to justify each step of a proof. Proofs cont.

## Geometric Proofs (cont.)

### General Tips for Writing Geometric Proofs:

• Draw a diagram and mark it with the given information.
• Write the hypothesis to be proven.
• Number each step.
• Statements with the same reason can be written separately or combined into one step.
• Every logical statement needs a reason! The reason can be given information, definition, theorems, and postulates.

### Avoiding Errors

Be careful when interpreting diagrams. There are some things you can conclude and some that you cannot.

### Can’t assume:

•  Collinear points

•  Measures of segments

•  Coplanar points

•  Measures of angles

•  Straight angles and lines

•  Congruent segments

•  Congruent angles

•   Linear pairs of angles

•  Right angles

•  Vertical angles

### Direct Proof

A direct geometric proof is a proof where you use deductive reasoning to make logical steps from the hypothesis to the conclusion. Each logical step needs to be justified with a reason.

There are several types of direct proofs:

• Two-column proof: Numbered statements go on the left side and the corresponding reasons go on the right side.
• Flowchart proof: Boxes and arrows are used to show the structure of the proof.
• Arrows show the flow of the logical argument, and the reasons are written below the statements they justify.
• Paragraph proof: The steps of the proof and their corresponding reasons are written as sentences in a paragraph.
• Transitional words such as so, then, and therefore help to make the logic clear.
• Coordinate proof: Place a geometric figure in a coordinate plane. Use variables to represent the coordinates of the figure so that the conclusion can be generalized.

### Two-Column Proof

A two-column proof is one way to write a geometric proof. Starting from GIVEN information, use deductive reasoning to reach the conjecture you want to PROVE.

• GIVEN: the hypothesis
• PROVE: the conclusion

Example of a Two-Column Proof:

### Reasons

1. Hypothesis

1. Given

2.

2.

...

...

n. Conclusion

n.

• The number of statements n will vary.
• Don’t forget to give a reason for the last statement!

### Indirect Proof

An indirect proof is where we prove a statement by first  assuming  that  it’s  false  and  then  proving  that  it’s impossible for the statement to be false (usually because it would lead to a contradiction). If the statement cannot be false, then it must be true.

Steps to write an indirect proof:

• Assume the opposite of the conclusion
• Continue to reason logically until a contradiction is reached.
• Since there is a contradiction, the original statement is true.

Use  variables  instead  of  specific  examples  so  that  the contradiction can be generalized.