Properties of Algebra

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

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

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

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

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

10. Properties of Equality Transitive property of equality If a = b and b = c, then a = c. 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. YES Didn’t we say this before?

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

13. Properties of Congruence Transitive property of Congruence 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. This is the same as in equality.

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

15. Recognition of Properties 1 If AB = CD , then Definition of congruent segments. 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. If then AB = CD. Definition of congruent segments.

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

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

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

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

20. Recognition of Properties 6 If AB = 30 and A = 5 , then 5B = 30. Substitution property of equality. If B is the midpoint of , then AB = BC. 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 Solving a first degree equation with 1 variable. 7x + 14 = 35 14 = 14 7x = 21 7 = 7 x = 3

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

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

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

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

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

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

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 Addition Subtraction Multiplication Division Each one is known as the property of equality.

29. Summary 3 The distributive property involves parentheses. a( b + c ) = ab + ac Multiplication is distributed to each item inside the parentheses. 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.