Deductive Reasoning “The proof is in the pudding.”“Indubitably.” Je solve le crime. Pompt de pompt pompt." Le pompt de pompt le solve de crime!" 2.6 Planning A Proof

1. Statement of the theorem 2. A diagram that illustrates the given information 4. Label diagram of what you are trying to prove with a question mark. 3. Label diagram of easily obtained data and the reasons for the data. 5. A series of statements and reasons linked together that leads from the given to the conclusion or to prove.

Sometimes you are given a diagram, the given, and to prove. At other times you will have to provide them yourself. This is why it is so important to be able to draw diagrams and to pick out the hypothesis and the conclusion.

When you draw the diagram try to avoid special cases. If you are to draw an angle don’t make it a right or straight angle. If you are to draw 2 intersecting lines, don’t make them perpendicular. Special cases will imply information that is not part of the given. It will lead you astray.

Before you start writing the 2 column proof, you need to plans what path will take to reach the conclusion. Sometimes you will see the path immediately. If so proceed immediately. If you cannot see the path to the conclusion, try different lines of thought. If you are stuck, then you can try the following approach.

What to do when stuck. 1. Relax 2. Write the given down. 3.Find facts that can be obtained from the given and the diagram using previous definitions, postulates and theorem. 4.Eventually, this might lead to the conclusion.

What to do when really stuck. 1. Make a list of items that would prove the conclusion. 2. Temporarily assume one of them is correct. 3. Ask what would be necessary to prove that assumed statement. 4. If you can find it from the diagram or from steps written downward from the given, then you are done.

This approach is known as the sandwich approach. You work from the top down and … You work from the bottom up. Soon the gap becomes so close that you can see the missing steps. The bottom and the top are the slices of bread. The final steps are the meat.

Words of Caution Proofs require time and experience. You will learn the methods. But the mastery will start to come easily in the next unit on parallel lines

Major Theorem Supplements of congruent angles are congruent. SCARC Exploration acdb ?? Since, then each angle needs the same amount to get to

Revisualized Supplements of congruent angles are congruent. SCARC Exploration a c db ? ? Since, then each angle needs the same amount to get to Understand ?

Note that the angles do not have to be connected. They do not have to be facing the same direction. d a cb ??

Supplements of congruent angles are congruent. SCARC ac db ? ? Given: To Prove: Strategy Find the values of and. If the values of and are the same, then they are congruent.

Before we begin, let’s review the steps to a proof. 1. Statement of the theorem 2. A diagram that illustrates the given information 4. Label diagram of what you are trying to prove with a question mark. 3. Label diagram of easily obtained data and the reasons for the data. 5. A series of statements and reasons linked together that leads from the given to the conclusion or to prove.

Supplements of congruent angles are congruent. ac db ? ? Given: To Prove: StatementsReasons Given Angle Add. Post. Subtr. Prop. Of Eq. Substitution Def. of congruent angles. g g Find value of angles Find equations with Angles C and D

Do you understand the flow of the proof ? 1. Find equations. 2. Manipulate the equations. 3. Satisfy the defintion of congruent angles. Do you appreciate the proof ? That may be a lot to ask for at this time !

2 nd Major Theorem Complements of congruent angles are congruent. CCARC Exploration a cd b ?? Since, then each angle needs the same amount to get to 90 0.

Complements of congruent angles are congruent. CCARC Exploration a cd b ?? Since, then each angle needs the same amount to get to Revisualized Understand ?

Note that the angles do not have to be connected. They do not have to be facing the same direction. d a c b h f g e

Complements of congruent angles are congruent. CCARC Given: To Prove: Strategy Find the values of and. If the values of and are the same, then they are congruent. a c ? d b ?

Before we begin, let’s review the steps to a proof. 1. Statement of the theorem 2. A diagram that illustrates the given information 4. Label diagram of what you are trying to prove with a question mark. 3. Label diagram of easily obtained data and the reasons for the data. 5. A series of statements and reasons linked together that leads from the given to the conclusion or to prove.

Complements of congruent angles are congruent. Given: To Prove: StatementsReasons Given Angle Add. Post. Subtr. Prop. Of Eq. Substitution Def. of congruent angles. Find value of angles Find equations with Angles C and D a c ? d b ?

Do you understand the flow of the proof ? 1. Find equations. 2. Manipulate the equations. 3. Satisfy the defintion of congruent angles. Do you appreciate the proof ? That may be a lot to ask for at this time !

Sample Problems 60

65

22

C’est fini. Good day and good luck. C’est fini. Good day and good luck.