review properties of equality and use them to write algebraic proofs
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Objectives. Review properties of equality and use them to write algebraic proofs. A proof uses logic, definitions, properties, and previously proven statements to “prove” that a conclusion is true. An important part of writing a proof is giving justifications to show that every step is valid. - PowerPoint PPT PresentationTRANSCRIPT
Holt McDougal Geometry
2-5 Algebraic Proof
Review properties of equality and use them to write algebraic proofs.
Objectives
Holt McDougal Geometry
2-5 Algebraic Proof
A proof uses logic, definitions, properties, and previously proven statements to “prove” that a conclusion is true.
An important part of writing a proof is giving justifications to show that every step is valid.
Possible ways to “prove” or justify each step:
1.A definition
2.A postulate
3.A property
4.Something given in the problem
Holt McDougal Geometry
2-5 Algebraic Proof
Holt McDougal Geometry
2-5 Algebraic Proof
The Distributive Property states that
a(b + c) = ab + ac.
Remember!
Holt McDougal Geometry
2-5 Algebraic Proof
Solve the equation 4m – 8 = –12. Write a justification for each step.
Example 1: Solving an Equation in Algebra
4m – 8 = –12
+8 +8
4m = –4
m = –1
Holt McDougal Geometry
2-5 Algebraic Proof
Check It Out! Example 1
t = –14
Solve the equation . Write a justification for each step.
Holt McDougal Geometry
2-5 Algebraic Proof
Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry.
A B
AB represents the length AB, so you can think of AB as a variable representing a number.
Helpful Hint
Holt McDougal Geometry
2-5 Algebraic Proof
Write a justification for each step.
Example 2: Solving an Equation in Geometry
NO = NM + MO
4x – 4 = 2x + (3x – 9)
4x – 4 = 5x – 9
–4 = x – 9
5 = x
Holt McDougal Geometry
2-5 Algebraic Proof
Check It Out! Example 2
Write a justification for each step.
x = 11
8x° = (3x + 5)° + (6x – 16)°
8x = 9x – 11
–x = –11
mABC = mABD + mDBC
Holt McDougal Geometry
2-5 Algebraic Proof
Check It Out! Example 2
Write a justification for each step.
x = 11
8x° = (3x + 5)° + (6x – 16)°
8x = 9x – 11
–x = –11
mABC = mABD + mDBC
Holt McDougal Geometry
2-5 Algebraic Proof
Classwork:
Exit Ticket with Partner
Homework:
•2.5 Practice B Homework Handout
• Must write out justification for each step to get credit!!
Holt McDougal Geometry
2-5 Algebraic Proof
Lesson Quiz: Part I
Solve each equation. Write a justification for each step.
1.
z – 5 = –12 Mult. Prop. of =
z = –7 Add. Prop. of =
Given
Holt McDougal Geometry
2-5 Algebraic Proof
Lesson Quiz: Part I
Solve each equation. Write a justification for each step.
1.
z – 5 = –12 Mult. Prop. of =
z = –7 Add. Prop. of =
Given
Holt McDougal Geometry
2-5 Algebraic Proof
Lesson Quiz: Part II
Solve each equation. Write a justification for each step.
2. 6r – 3 = –2(r + 1)
Given
6r – 3 = –2r – 2
8r – 3 = –2
Distrib. Prop.
Add. Prop. of =
6r – 3 = –2(r + 1)
8r = 1 Add. Prop. of =
Div. Prop. of =
Holt McDougal Geometry
2-5 Algebraic Proof
WARM UP
Give an example of the following properties:
1. Reflexive Property of Equality:
2. Symmetric Property of Equality:
3. Transitive Property of Equality:
Holt McDougal Geometry
2-5 Algebraic Proof
Objective:
Identify properties of equality and congruence.
Remember…..segments with equal lengths are congruent and that angles with equal measures are congruent.
So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence.
Holt McDougal Geometry
2-5 Algebraic Proof
Holt McDougal Geometry
2-5 Algebraic Proof
Numbers are equal (=) and figures are congruent ().
Remember!
Holt McDougal Geometry
2-5 Algebraic Proof
Identify the property that justifies each statement.
A. QRS QRS
B. m1 = m2 so m2 = m1
C. AB CD and CD EF, so AB EF.
D. 32° = 32°
Example 1: Identifying Property of Equality and Congruence
Symm. Prop. of =
Trans. Prop of
Reflex. Prop. of =
Reflex. Prop. of .
Holt McDougal Geometry
2-5 Algebraic Proof
Check It Out! Example 2
Identify the property that justifies each statement.
A. DE = GH, so GH = DE.
B. 94° = 94°
C. 0 = a, and a = x. So 0 = x.
D. A Y, so Y A
Sym. Prop. of =
Reflex. Prop. of =
Trans. Prop. of =
Sym. Prop. of
Holt McDougal Geometry
2-5 Algebraic Proof
Lesson Review
Identify the property that justifies each statement.
1. x = y and y = z, so x = z.
2. DEF DEF
3. AB CD, so CD AB.
Trans. Prop. of =
Reflex. Prop. of
Sym. Prop. of
Holt McDougal Geometry
2-5 Algebraic Proof
Property Equality Example Congruence Example
Reflexive
Symmetric
Transitive
Classwork/Homework
come up with a unique example of each