Introduction to Geometry Proofs

Proof Vocabulary

Postulate Theorem

Postulate: Rules that are accepted without proof

Theorem: A true statement that follows as a result of other true statements.

Logical Argument in Algebra

Given x + y = 60 Given x = 5 Prove y = 55

Use your algebra knowledge to write a proof. Justify each step you write.

Algebra Proof Solution

Follow the steps.

x + y = 60 x = 5 5 + y = 60

y = 55

Justify the steps.

Given Given Substitution

Property of Equality

Subtraction Property of Equality

Types of Geometry Proof

Two Column Proofs This third example is the most commonly used

type of proof. We will focus on this type of proof in class.

Paragraph Proofs Find an example in your textbook and read it

to your table partner.

Flow Chart Proofs Find an example in your textbook and copy the

steps into your Geometry notebook.

Two Column Proofs

Statements

In this column we write the logical steps that lead us to the end result.

Reasons

For each statement, we must use a postulate or theorem that supports the statement.

Two Column ProofFill in the blanks to complete the proof of

the Reflexive Property of the Congruence of Angles.

Statements

A is an angle. Measure of A = Measure of

A Angle A is congruent to

Angle A

Reasons

______________________ ______________________ ______________________

Two Column Proof Check your solution for the proof of the Reflexive Property of the Congruence of

Angles. Statements

A is an angle. Measure of A = Measure of

A Angle A is congruent to

Angle A

Reasons

Given Reflexive Property of

Equality Definition of Congruent

Angles

Another 2 Column Proof n m2 3 1

Complete the following proof by filling in the blanks.

Given: Angle 1 and Angle 2 are supplementaryProve: n is parallel to m

Statements Reasons1) Angle 1 and Angle 2 are supplementary. 1)______________________2) Angle 1 and Angle 3 are a linear pair. 2)______________________3)_____________________________ 3) Linear Pair Postulate4)_____________________________ 4) Congruent Supplements

Theorem5) n is parallel to m. 5) ______________________

One last 2 Column Proof n m2 3 1

Check your work to see how well you are doing.

Given: Angle 1 and Angle 2 are supplementaryProve: n is parallel to m

Statements Reasons1) Angle 1 and Angle 2 are supplementary. 1) Given2) Angle 1 and Angle 3 are a linear pair. 2) Definition of Linear Pair3) Angle 1 and Angle 3 are supplementary. 3) Linear Pair Postulate4) Angle 2 is congruent to Angle 3 4) Congruent Supplements Theorem5) n is parallel to m. 5) Corresponding Angles

Converse

Paragraph Proof

See page 122 (Middle section of page) “Paragraph Proof”

A proof that can be written in paragraph form is called a paragraph proof.

See example on bottom of page 122

Flow Chart Proofsj

5 6

k

Given: angle 5 is congruent to angle 6, angle 5 and 6 are a linear pair.

Prove: j is perpendicular to k.

Put the following statements in the proper order to complete the proof. When you have finished, compare your solution to your partners.

angle 5 is congruent to angle 6

j is perpendicular to k measure of 5 = 90°

measure of 5 = measure of 6angles 5 and 6 are a linear pair.

2(measure of 5) = 180°

measure of 5 + measure of 6 = 180°

angle 5 and angle 6 are supplementary

angle 5 is a right angle

measure of 5 + measure of 5 = 180°

Flow Chart Proofsj

5 6

k

Given: angle 5 is congruent to angle 6, angle 5 and 6 are a linear pair.

Prove: j is perpendicular to k.

Now that you have the statements in a logical order, add a reason to each statement. Reasons are based on properties, postulates and theorems.

When you have finished, bring your paper to the teacher. You will be asked to explain your reasoning.