1. Properties of Algebra There are various properties from algebra that allow us to perform certain tasks. We review them now to refresh your memory on the process and terminology. We will also add a few new properties which you might not be familiar.

2. 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!"

3. Properties of Equality If a = b and c = d, then a + c = b + d. Addition property of equality restated If a = b and c = d, then a + c = b + d. Example a = b 3 = 3 a + 3 = b + 3 Euclid referred to this property as… “Equals when added to equals are equal.”

4. Properties of Equality If a = b and c = d, then a - c = b - d. Subtraction property of equality restated If a = b and c = d, then a - c = b - d. Example a = b 3 = 3 a - 3 = b - 3 Euclid referred to this property as… “Equals when subtracted from equals are equal.”

5. Properties of Equality If a = b and c = d, then ac = bd. Multiplication property of equality restated If a = b and c = d, then ac = bd. Example a = b 3 = 3 3a = 3b Euclid referred to this property as… “Equals when multiplied by equals are equal.”

6. Properties of Equality If a = b and, then Division property of equality restated If a = b and c = c, then Example a = b 3 = 3 Euclid referred to this property as… “Equals when divided by equals are equal.”

7. Properties of Equality If a = b and, then Division property of equality restated If a = b and c = c, then Euclid referred to this property as… “Equals when divided by equals are equal.” Why must c not equal zero? You are not allowed to divide by zero. Numbers divided by zero are undefined.

8. Properties of Equality a = a Reflexive property of equality This is really obvious. Nevertheless, it needs a name. When you look into a mirror, you see your reflection. Think of the equal sign as a mirror. It might help you remember the term.

9. Properties of Equality If a = b, then b = a. Symmetric property of equality This is really obvious. Nevertheless, it needs a name.

10. Properties of Equality If a = b and b = c, then a = c. Transitive property of equality This is really obvious. Nevertheless, it needs a name. It might be helpful to associate this concept with traveling from LA to NYC with a stop over at Chicago. The transfer of planes allows you to reach your final destination.

11. Properties of Congruence Reflexive property of congruence This is really obvious. Nevertheless, it needs a name. When you look into a mirror, you see your reflection. Think of the equal sign as a mirror. It might help you remember the term. Didn’t we say this before? YES

12. Properties of Congruence If, then. Symmetric property of Congurence This is really obvious. Nevertheless, it needs a name. If, then. Symmetric is the same for equality and congruence.

13. Properties of Congruence Transitive property of Congruence This is the same as in equality. It might be helpful to associate this concept with traveling from LA to NYC with a stop over at Chicago. The transfer of planes allows you to reach your final destination.

14. Distributive Property a( b + c) = ab +ac Implied multiplication

15. Recognition of Properties 1 If AB = CD, then Definition of congruent segments. If then AB = CD. If a = b and b = c, then a = c. Transitive property of equality. If a + b = 10 and b = 3, then a + 3 = 10. Substitution property of equality. Definition of congruent segments.

16. Recognition of Properties 2 If a = b and x = y, then a + x = b + y. If a = b and x = y, then a - x = b - y. If a = 7, then a + 3 = 10. Addition property of equality. Why? I added 3 to both sides. Remember Euclid? “Equals when added to equals are equal.” Subtraction property of equality. +3 -x -y +x +y

17. Recognition of Properties 3 If B is on line AC and AB = BC, then b is the midpoint of If A = B, then A + 3 = B + 3. +3 Addition property of equality. Definition of midpoint. Symmetric property of equality. switch sides Reflexive property of equality. Mirror image

18. 11( 4x + 7) = 44x + 77 Recognition of Properties 4 If a = b and b = c and c = 11, then a = 11. If a = 11, then a – 3 = 8. If a = b and c = 12, then Distributive property. Transitive property of equality. -3 c c Subtraction property of equality. Division property of equality.

19. Recognition of Properties 5 If, then a = 42. If 8x = 48, then x = 6. If 2y – 7 = 11, then 2y = 18. If then Multiplication property of equality. Division property of equality. Addition property of equality. Symmetric property of equality. switch sides 7 __ ___ 8 +7

20. Recognition of Properties 6 If B is the midpoint of, then AB = BC. If AB = 30 and A = 5, then 5B = 30. Substitution property of equality. Definition of midpoint.

21. Proofs You have been doing proofs all along in Algebra I. When? When you solved equations, you were actually doing proofs – algebraic proofs. The major difference between equations and geometric proofs is in the form. 7( x + 2 ) = 35 7x + 14 = 35 14 = 14 7x = 21 7 = 7 x = 3 Solving a first degree equation with 1 variable.

22. Proofs The major difference between equations and geometric proofs is in the form. 7( x + 2 ) = 35 7x + 14 = 35 14 = 14 7x = 21 7 = 7 x = 3 If 7( x + 2 ) = 35, then x =3. Statements Reasons Given Information Distributive Property Reflexive Property Subtraction Prop. Of Equality Reflexive Property Division Prop. Of Equality The only difference is that the reasons/justification for each step must be written in geometry. Written as a conditional.

23. 3x = 4( 7 – x ) 7 = 7 4x = 4x 4 = 4 x = 4 StatementsReasons Given Distributive Property Reflexive Property Multiplication Prop. Of Equality Reflexive Property Division Prop. Of Equality Start Finish 3x = 28 – 4x 7x = 28 Addition Prop. Of Equality Reflexive Property Note this is a lot of writing. You will need To abbreviate

24. 3x = 4( 7 – x ) 7 = 7 4x = 4x 4 = 4 x = 4 StatementsReasons Given Distr. Prop. Of = Reflexive Prop. Mult. Prop. Of = Reflexive Prop. Div. Prop. Of = Start Finish 3x = 28 – 4x 7x = 28 + Prop. Of = Reflexive Prop. This is a lot less writing.

25. 3x = 4( 7 – x ) 7 = 7 4x = 4x 4 = 4 x = 4 StatementsReasons Given Distr. Prop. Of = Reflexive Prop. Mult. Prop. Of = Reflexive Prop. Div. Prop. Of = Start Finish 3x = 28 – 4x 7x = 28 + Prop. Of = Reflexive Prop. In algebra, certain easy steps are left out, because they are understood. Eventually, we will do the same. But not just yet! Generally in algebra the reflexive steps are invisible or left out for speed and/or convenience.

26. Geometric Proof 1 If AB = CD, then AC = BD. ADCB Given: AB = CD Prove: AC = BD First step is to label the diagram. ? ? gg StatementsReasons AB = CD BC = BC AB+BC = BC+CD AB+BC = AC BC+CD = BD AC = BD Given Reflexive Prop. + Prop. Of = Seg. Addition Post. Seg. + Post. Substitution Labeling means marking and giving the reasons next to the markings. Start with given and then add steps to reach the conclusion.

27. g Geometric Proof 2 If AB = BE and DB = CB, then AC = DE. A D C B Given: AB = BE DB = CB Prove: AC = BD 1st step is to label the diagram. ? ? gg StatementsReasons AB = BE BC = DB AB+BC = DB+BE AB+BC = AC DB+BE = DE AC = DE Given + Prop. Of = Seg. Addition Post. Seg. + Post. Substitution Labeling means marking and giving the reasons next to the markings. Start with given and then add steps to reach the conclusion. E g Given

28. Summary 1 The properties of algebra are used as reasons or justifications of steps in proofs. 2 Four of the properties are associated with arithmetic operations in equations Euclid said it simply as: Equals when by equals are equal. Added Subtracted Multiplied Divided Each one is known as the property of equality. Addition Subtraction Multiplication Division

29. Summary 3 The distributive property involves parentheses. Multiplication is distributed to each item inside the parentheses. a( b + c ) = ab + ac 4 Proofs are a process of linking statement together from the hypotheses to the conclusion. It will take over a month to get comfortable with the process of writing proofs. Relax. Be patient. (hard to do) It WILL come.

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