Part of this course is to help improve deductive reasoning skills and one method that will be utilized is proof writing. Here are a few basics to help begin writing proofs.

How to prove the conditional statement:  If  P (hypothesis), then Q (conclusion).

Writing Proofs: A proof is generally written as a sequence of statements with a reason given for each statement.  You MUST have a valid reason for each statement that you make. The reason shows that you know why the statement that you wrote is true – i.e. you can justify your argument.

The Two-Column Format:  We will begin with “two-column proofs” to help keep statements and reasons organized. After some practice, we will then do some “formal proofs”, where the statements and reasons are written as complete sentences and the entire proof is written in paragraph form.

Two-column format:

Strategy Guidelines with example:

Example: Prove that if a triangle has a pair of complementary interior angles, then the triangle is a right triangle.

1. As with any problem, you must first be able to identify what information is given  (The Hypothesis), and what it is you are trying to show (The Conclusion). 

            Given:  A triangle with a pair of complementary interior angles

            Prove:  The triangle is a right triangle

    2.   Draw a picture from the given information only and label parts appropriately.

 3.  Repeat step (1) using your picture.

            ^{Given:  with    and complementary.}

            ^{Prove: is a right triangle}.

    4.  Be sure that you know the definitions of the words in both the Given  and            Prove  portions. Write them down.

            Given:    ^{and complementary angles means .}

            Prove: A right triangle is a triangle with one right angle (an angle measuring 90 degrees). In this case it must be shown that angle B is a right angle.

    5.  Write down any other information you know that may help you prove what   you are trying to prove. These may be Postulates, Propositions, Theorems,  or Corollaries that we have discussed in class or you have read in your text.

            Example: The sum of the interior angles of a triangle is 180 degrees.

    6.  Determine if you can see why the part you are trying to prove is true from the previous steps. Talk yourself through it first. If not, go back to step (5).

            Example: "Since angles A, B and C sum to 180 degrees and angles A  and C sum to 90 degrees, angle B must be a 90 degree, or right, angle."      

    7.  Now you can start writing the proof.

Statement	Reason
and complementary.	Given
	Definition of complementary angles
	The sum of the interior angles of a triangle is 180 degrees
	Substitution (I below)
	Subtraction property of equality (I below)
is a right triangle	A triangle with a right angle is a right .

Below are some more familiar reasons that are commonly used. 

What do you use for reasons? (Please see Appendix 1 and Appendix 2 and be sure you know how to use them)

1)    The hypothesis (P) - the given information is usually the first statement. The reason is that this information was given. (See table above)

2)    Definitions can always be used as reasons.

3)    Postulates can always be used as reasons. 

4)    Previously proven theorems and corollaries can be used as reasons.

5)    Later in the semester we may need to use a construction to help in writing a proof.

6)     Other reasons come from properties of numbers and geometric shapes, and these are listed below.

 Let a, b, c, x and  y be real numbers. Let X, Y, Z be some geometric objects – like segments, angles, or triangles.

I)      Properties of equality:                                                                 REASON

    Addition:                If a=b, then a+c = b+c                          (The addition property of equality)

    Subtraction:           If a=b then a-c = b – c                           (The subtraction property of equality)

    Multiplication:       If a=b then ac = bc for c ¹ 0                 (The multiplication property of equality)

    Division:                 If a=b then a/c = b/c for c ¹ 0               (The division property of equality)

    Substitution:          If x + a = b and y = x then y + a = b      (Substitution)

II)      Properties of Inequality                                                                              REASON

Addition:              If a>b, then a+c > b+c               (The addition property of inequality)

Subtraction:         If a>b then a-c > b – c               (The subtraction property of inequality)

Multiplication:    If a>b then ac > bc for c > 0      (The multiplication property of inequality)

Division:              If a>b then a/c > b/c for c > 0     (The division property of inequality)

III)  More properties of equality of numbers.                                               REASON

Reflexive:                  a=a                                                        (Reflexive property of equality)

Symmetric:                If a = b, then b = a                              (Symmetric property of equality)

Transitive:                If a = b and b = c, then a = c              (Transitive property of equality)

IV)      Properties of congruence.                                                                         REASON

Reflexive:                 X @ X                                               (Reflexive property of congruence)

Symmetric:                If X @ Y then Y @ X                       (Symmetric property of congruence)

Transitive:                If X @ Y and Y @ Z then X @ Z     (Transitive property of congruence)