GEOMETRY. 2.4 Algebraic Reasoning

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2 GEOMETRY 2.4 Algebraic Reasoning

3 2.4 ESSENTIAL QUESTION How can algebraic properties help you solve an equation?

4 GOALS Use properties from algebra. Use properties to justify statements.

5 ADDITION PROPERTY Addition Property If a = b, then a + c = b + c. Example x 12 = 15 Given x = x = 27 Add. Prop.

6 SUBTRACTION PROPERTY Subtraction Property If a = b, then a c = b c Example x + 30 = 45 Given x = x = 15 Subtr. Prop.

7 MULTIPLICATION PROPERTY Multiplication Property If a = b, then ac = bc Example 2 3 x 12 Given x (12) x 18 Mult. Prop.

8 DIVISION PROPERTY Division Property If a b, then a b c c. (c 0) Example 12x 2.4 Given 12x x 0.2 Divis. Prop.

9 SUBSTITUTION PROPERTY Substitution Property If a = b, then a can be substituted for b (or b for a) in any equation or expression. Example x = y Given x 7 = 5 y 7 = 5 Given Subst. Prop.

10 DISTRIBUTIVE PROPERTY Distributive Property a(b + c) = ab + ac Example 5(x 3) 5x 15 Given Distrib. Prop

11 EXAMPLE 1 Using the algebraic properties, solve the equation below and justify each step. 3x + 2 = 23 4x 3x x = 23 4x + 4x 7x + 2 = 23 7x = x = 21 x = 3 Given Addition Property Simplify Subtraction Property Simplify Division Property

12 EXAMPLE 2 Using the algebraic properties, solve the equation below and justify each step. -5( 7w + 8) = 30 Given -35w 40 = 30-35w = 70 w = -2 Distributive Property Addition Property Division Property

13 EXAMPLE 3 Solve x+2(x 3) = 5x + 2 x + 2(x 3) = 5x + 2 x + 2x 6 = 5x + 2 3x 6 = 5x + 2 3x = 5x + 8 2x = 8 x = 4 Given Distributive Property Simplify Addition Property Subtraction Property Division Property

14 MORE PROPERTIES Algebraic Properties Geometric Properties Real Numbers Segments Angles Reflexive a = a AB AB A A Symmetric If a = b, then b = a Transitive If a = b, and b = c, then a = c If AB CD, then CD AB If AB CD, and CD EF, then AB EF If A B, then B A If A B, and B C, then A C

15 IDENTIFY THE PROPERTY EXAMPLE 4 If 43 = x, then x = 43. Property? Symmetric

16 IDENTIFY THE PROPERTY EXAMPLE 5 If 3x = 12, then 12x = 48 Property? Multiplication

17 IDENTIFY THE PROPERTY EXAMPLE 6 If x = y, and y = 10, then x = 10 Property? Transitive

18 IDENTIFY THE PROPERTY Your turn If x = 12, then x + 2 = 14 Property? Addition

19 USING A PROPERTY Example 7 Addition Property: 16 If n = 14, then n + 2 =

20 USING A PROPERTY Example 8 Symmetric: If AB = CD, then CD = AB

21 USING A PROPERTY Your Turn Transitive: If m A = m B, and m B = m C, then m A =. m C

22 JUSTIFICATION One of the main reasons to study Geometry is to learn how to prove things. The whole business of math is proving things. To prove things in math you must be able to justify everything with legitimate reasons. Our reasons include: postulates, definitions and algebra properties.

23 Before we do a proof Any geometry proof must begin with information that we know is true. This will be given to us as a place to start, so it is called the given. There can be one or more givens in a problem.

24 USING GEOMETRY PROPERTIES Example 9 Given: AC = BD. Prove that AB = CD. A B C D AC = BD AB + BC = AC BC + CD = BD AB + BC = BC + CD AB = CD Given Seg. Add. Post Seg. Add. Post Substitution Subtraction

25 THIS IS PROOF. 1 st hour If you feel uncomfortable and confused, that s normal. Everyone is confused with proof at first. There is only one way to learn proof: PRACTICE. You have to know the properties, postulates and definitions. You must diligently practice by doing the homework every night NO EXCUSES. You learn by making mistakes. Everyone does.

26 PROOF IS ESSENTIAL. Proof is a mandatory part of higher math. If you plan on going to college and/or taking more advanced math you must prove things. Geometry is the place to learn to do this. We will do proofs until the end of the year. Don t fight it, they are not going to go away.

27 WHAT HAVE YOU LEARNED? Example 10 Transitive Addition Subtraction Reflexive Substitution Multiplication Symmetric

28 NOW TRY THESE 15 ½ C AF m DCF = m MJC DB 15 YZ = JK

29 PROBLEM 11 Fill in the reasons for each step. 3(2x 4) = 5x + 2 Given 6x 12 = 5x + 2 a. Distributive x 12 = 2 b. Subtraction x = 14 c. Addition

30 YOUR TURN Fill in the reasons for each step. 4x + 8 = 2x 12 2x + 8 = -12 2x = -20 x = -10 Given a. Subtraction b. Subtraction c. Division

31 ASSIGNMENT