Consider this quick example. Let p be the statement 'Ice is cold' and let q be the statement '5 is greater than three + four'. Is p and q true? Because q is false, p and q is also false. It doesn't matter if p is false. It only takes one of the statements to be false in order for the conjunction to be false. To Prove It is essential to determine what you know and what is to be proven in order to have a correct proof. For many proof exercises, this information is given just as it is represented in this graphic. If the statement to be proved is written as a conditional, the conclusion contains what is to be proven. If the statement is not written as a conditional, it can always be rewritten in If, Then format. This graphic also shows three of the main components in completing a two column proof. Two column proof formats provide an easy to see and easy to follow guide for completing any proof. The two column format enables us to show each conclusion arrived at and the reason that supports that conclusion. Notice the first two headings in the graphic are and “Given” “Prove". This theorem is already written as a conditional so the are found in the “givens” hypothesis and the is contained in the conclusion. The portion of “prove” “Given” the two column proof is parallel lines cut by a and the Prove “two third line” statement is interior angles are “alternate congruent”. The hypothesis becomes the and the conclusion of the sum of the “Given”, angles equals 180 degrees is what needs to be proven. Unit 2 Lesson 16 Proof Formats: The Plan of the Proof Proof Formats: The Plan of the Proof In this example, you are given a diagram of two supplementary angles. (The Angle-Addition-Postulate or AAP allows you to add the two angles and set them equal to 180 degrees.) Through algebraic properties of subtraction and division we can prove that x = 40. With a quick check, you can know that your answer of x = 40 satisfies the condition that the angles are supplementary. Always begin a proof with a plan in mind. Plan of the Proof
Consider this quick example. Let p be the statement 'Ice is cold' and let q be the statement '5 is greater than three + four'. Is p and q true? Because q is false, p and q is also false. It doesn't matter if p is false. It only takes one of the statements to be false in order for the conjunction to be false. To Prove It is essential to determine what you know and what is to be proven in order to have a correct proof. For many proof exercises, this information is given just as it is represented in this graphic. If the statement to be proved is written as a conditional, the conclusion contains what is to be proven. If the statement is not written as a conditional, it can always be rewritten in If, Then format. This graphic also shows three of the main components in completing a two column proof. Two column proof formats provide an easy to see and easy to follow guide for completing any proof. The two column format enables us to show each conclusion arrived at and the reason that supports that conclusion. Notice the first two headings in the graphic are “Given” and “Prove". This theorem is already written as a conditional so the “givens” are found in the hypothesis and the “prove” is contained in the conclusion. The “Given” portion of the two column proof is “two parallel lines cut by a third line” and the Prove statement is “alternate interior angles are congruent”. The hypothesis becomes the “Given”, and the conclusion of the sum of the angles equals 180 degrees is what needs to be proven. Unit 2 Lesson 16 Proof Formats: The Plan of the Proof Proof Formats: The Plan of the Proof In this example, you are given a diagram of two supplementary angles. (The Angle-Addition-Postulate or AAP allows you to add the two angles and set them equal to 180 degrees.) Through algebraic properties of subtraction and division we can prove that x = 40. With a quick check, you can know that your answer of x = 40 satisfies the condition that the angles are supplementary. Always begin a proof with a plan in mind. Plan of the Proof
week_5_geometry_notes.docx