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Algebraic Proofs and Properties
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Algebraic Proofs and Properties

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Questions and Answers

What is the use of an algebraic proof?

To justify the steps of an algebraic equation

What are the seven main algebraic proofs/properties?

The reflexive, symmetric, transitive, addition/subtraction, multiplication/division, substitution, and distribution properties

What does the reflexive property state?

For every number a, a = a

What does the symmetric property state?

<p>For all numbers a and b, if a = b, then b = a</p> Signup and view all the answers

What does the transitive property state?

<p>For all numbers a, b, and c, if a = b and b = c, then a = c</p> Signup and view all the answers

What does the addition and subtraction property state?

<p>For all numbers a, b, and c, if a = b, then a + c = b + c and a - c = b - c</p> Signup and view all the answers

What does the multiplication and division property state?

<p>For all numbers a, b, and c, if a = b, then ac = bc and if c doesn't equal 0, then a/c = b/c</p> Signup and view all the answers

What does the substitution property state?

<p>For all numbers a and b, if a = b, then a may be replaced by b in any equation or expression</p> Signup and view all the answers

What does the distribution property state?

<p>For all numbers a, b, and c, a(b + c) = ab + ac</p> Signup and view all the answers

List all of the algebraic steps and properties that prove each is valid in the following equation: 6x + 2(x-1) = 30

<p>6x + 2(x-1) = 30 [given], 6x + 2x - 2 = 30 [distributive property], 8x - 2 = 30 [substitution], 8x - 2 + 2 = 30 + 2 [addition property], 8x = 32 [substitution], 8x / 8 = 32 / 8 [division property], x = 4 [substitution]</p> Signup and view all the answers

List all of the algebraic steps and properties that prove each is valid in the following equation: 4x + 6/2 = 9

<p>4x + 6/2 = 9 [given], 2(4x + 6/2) = 9(2) [multiplication property], 4x + 6 = 18 [substitution], 4x + 6 - 6 = 18 - 6 [subtraction property], 4x = 12 [substitution], 4x/4 = 12/4 [division property], x = 3 [substitution]</p> Signup and view all the answers

List all of the algebraic steps and properties that prove each is valid in the following equation: 4x + 8 = x + 2

<p>4x + 8 = x + 2 [given], 4x + 8 - x = x - x + 2 [subtraction property], 3x + 8 = 2 [substitution], 3x - 8 = 2 - 8 [subtraction property], 3x = -6 [substitution], 3x/3 = -6/3 [division property], x = -2 [substitution]</p> Signup and view all the answers

What do geometric proofs deal with and what do they use to prove statements true?

<p>They deal with numbers as measures and use the properties of numbers to prove given statements true.</p> Signup and view all the answers

What are the three geometric proofs/properties?

<p>The reflexive, symmetric, and transitive properties</p> Signup and view all the answers

What does the reflexive property state about segments and angles?

<p>Segments: AB = AB; Angles: angle A = angle A</p> Signup and view all the answers

What does the symmetric property state about segments and angles?

<p>Segments: if AB = CD, then CD = AB; Angles: if angle A = angle B, then angle B = angle A</p> Signup and view all the answers

What does the transitive property state about segments and angles?

<p>Segments: if AB = CD and CD = EF, then AB = EF; Angles: if angle 1 = angle 2 and angle 2 = angle 3, then angle 1 = angle 3</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if angle 1 = angle 2, angle 2 = angle 3, then angle 1 = angle 3.

<p>angle 1 = angle 2 [given], angle 2 = angle 3 [given], angle 1 = angle 3 [transitive property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if angle 1 = angle 2, then angle 2 = angle 1.

<p>angle 1 = angle 2 [given], angle 2 = angle 1 [symmetric property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if angle 1 = 90 and angle 2 = angle 1, then angle 2 = 90.

<p>angle 1 = 90 [given], angle 2 = angle 1 [given], angle 2 = 90 [substitution property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if AB = RS and RS = WY, then AB = WY.

<p>AB = RS [given], RS = WY [given], AB = WY [transitive property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if AB = CD, then 1/2AB = 1/2CD.

<p>AB = CD [given], 1/2AB = 1/2CD [multiplication property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if angle 1 + angle 2 = 110 and angle 2 = angle 3, then angle 1 + angle 3 = 110.

<p>angle 1 + angle 2 = 110 [given], angle 2 = angle 3 [given], angle 1 + angle 3 = 110 [substitution property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: RS = RS.

<p>Reflexive property</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if AB = RS and TU = WY, then AB + TU = RS + WY.

<p>AB = RS [given], TU = WY [given], AB + TU = RS + WY [addition property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: if angle 1 = angle 2 and angle 2 = angle 3, then angle 1 = angle 3.

<p>angle 1 = angle 2 [given], angle 2 = angle 3 [given], angle 1 = angle 3 [transitive property]</p> Signup and view all the answers

Use geometric proofs to prove the following statement: A formula for the area of a triangle is A = 1/2bh. Prove that bh is equal to two times the area of the triangle.

<p>A = 1/2bh [given], (2)A = (2)1/2bh [multiplication], 2A = bh</p> Signup and view all the answers

Study Notes

Algebraic Proofs and Properties

  • Algebraic proofs are used to justify the steps involved in solving algebraic equations.
  • Main algebraic properties include:
    • Reflexive Property: a = a for any number a.
    • Symmetric Property: If a = b, then b = a for any numbers a and b.
    • Transitive Property: If a = b and b = c, then a = c for any numbers a, b, and c.
    • Addition/Subtraction Property: If a = b, then a + c = b + c and a - c = b - c.
    • Multiplication/Division Property: If a = b, then ac = bc; if c ≠ 0, then a/c = b/c.
    • Substitution Property: If a = b, then a can be replaced by b in any equation or expression.
    • Distribution Property: a(b + c) = ab + ac for any numbers a, b, and c.

Application of Algebraic Proofs

  • Example of proof for the equation 6x + 2(x-1) = 30:
    • Uses distributive, substitution, addition, and division properties to find x = 4.
  • Example of proof for the equation 4x + 6/2 = 9:
    • Involves multiplication, substitution, and division properties leading to x = 3.
  • Example of proof for the equation 4x + 8 = x + 2:
    • Applies subtraction and division properties to solve for x = -2.

Geometric Proofs

  • Geometric proofs involve numbers as measures, employing properties to establish truth in statements.
  • Key geometric properties mirror algebraic properties:
    • Reflexive Property: For segments or angles, AB = AB and angle A = angle A.
    • Symmetric Property: For segments or angles, if AB = CD, then CD = AB; if angle A = angle B, then angle B = angle A.
    • Transitive Property: For segments or angles, if AB = CD and CD = EF, then AB = EF; similarly for angles.

Applications of Geometric Proofs

  • To prove relationships between angles:
    • Example: If angle 1 = angle 2 and angle 2 = angle 3, then angle 1 = angle 3 via transitive property.
    • Example: If angle 1 = 90 and angle 2 = angle 1, then angle 2 = 90 through substitution.
  • To establish equivalence of segments:
    • Example: If AB = CD, then 1/2AB = 1/2CD employs the multiplication property.

Formula and Area Proofs

  • For triangles, the area formula A = 1/2bh leads to proving that bh equals two times the area:
    • Start with A = 1/2bh and multiply both sides by 2 to derive 2A = bh.

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Description

This quiz explores key algebraic proofs and properties fundamental to solving equations. Topics include various properties such as reflexive, symmetric, and transitive. Apply these concepts to validate your understanding with practical examples.

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