Geometry

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 Name

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.

Geometric Proofs

Geometry

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 assume:

Can’t assume:

•  Collinear points

•  Measures of segments

•  Coplanar points

•  Measures of angles

•  Straight angles and lines

•  Congruent segments

•   Adjacent angles

•  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:

    Statements  

    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.