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April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 2
Goals Use properties from algebra. Use properties to justify
statements.
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 3
Algebra Properties Don’t copy all of this down You have had most before. Copy
the ones that are new to you You need to have them all
memorized. This as well as all presentations
are available on-line.
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 4
Addition PropertyAddition Property
If a = b, then a + c = b + c.
Example
x – 12 = 15
x – 12 + 12 = 15 + 12
x = 27
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 5
Subtraction PropertySubtraction PropertyIf a = b, then a – c = b – c
Example
x + 30 = 45
x + 30 – 30 = 45 – 30
x = 15
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 6
Multiplication PropertyMultiplication Property
If a = b, then ac = bc
Example 2 123
3 2 3 (12)2 3 2
18
x
x
x
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 7
Division Property
0)(c cb
ca
then b, a If .
Division Property
Example
2.012
4.2
12
12
4.212
x
x
x
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 8
Other Properties
Reflexive Property For any number a, a = a.
Symmetric Property If a = b, then b = a.
Transitive Property If a = b and b = c, then a = c
Substitution Property If a = b and a = c, then b = c
Distributive Property a(b + c) = ab + ac
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 9
Identify the Property
If 43 = x, then x = 43.
Property?
Symmetric
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 10
Identify the Property
If 3x = 12, then 12x = 48
Property?
Multiplication
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 11
Identify the Property
If x = y, and y = 10, then x = 10
Property?
Transitive
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 12
Identify the Property
If x = 12, then x + 2 = 14
Property?
Addition
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 13
Using a Property
Addition Property:
If n = 14, then n + 2 = _________16
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 14
Using a Property
Symmetric:
If AB = CD, then CD = ________ AB
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 15
Using a Property
Transitive:
If mA = mB, and mB = mC,
then mA = _________.mC
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 16
Justification
One of the main reasons to study Algebra 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.
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 17
Example
Solve 3x + 12 = 8x – 18 and write a reason for each step.3x + 12 = 8x – 18 Given
–5x = –30
Subtraction Property
x = 6
Division Property
3x – 8x + 12 – 12 = 8x – 8x – 18 – 12
Simplify (combine like terms)–5x/(–5) = –30/(–
5)
Simplify
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 18
Another Example (w/o intermediate steps)
Solve x+2(x – 3) = 5x + 2x + 2(x – 3) = 5x + 2x + 2x – 6 = 5x + 2
3x – 6 = 5x + 2
3x = 5x + 8
–2x = 8
x = – 4
Given
Distributive Prop.
Simplify
Addition Prop.
Subtraction Prop.
Division Prop.
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 19
Before we do a proof…
Any algebra 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.
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 20
This is “Proof”.
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.
April 10, 2023Geometry 2.4 Reasoning with Algebra
Properties 21
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. Algebra 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.