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?
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What does the transitive property state?
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What does the addition and subtraction property state?
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What does the multiplication and division property state?
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What does the substitution property state?
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What does the distribution property state?
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List all of the algebraic steps and properties that prove each is valid in the following equation: 6x + 2(x-1) = 30
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List all of the algebraic steps and properties that prove each is valid in the following equation: 4x + 6/2 = 9
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List all of the algebraic steps and properties that prove each is valid in the following equation: 4x + 8 = x + 2
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What do geometric proofs deal with and what do they use to prove statements true?
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What are the three geometric proofs/properties?
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What does the reflexive property state about segments and angles?
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What does the symmetric property state about segments and angles?
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What does the transitive property state about segments and angles?
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Use geometric proofs to prove the following statement: if angle 1 = angle 2, angle 2 = angle 3, then angle 1 = angle 3.
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Use geometric proofs to prove the following statement: if angle 1 = angle 2, then angle 2 = angle 1.
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Use geometric proofs to prove the following statement: if angle 1 = 90 and angle 2 = angle 1, then angle 2 = 90.
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Use geometric proofs to prove the following statement: if AB = RS and RS = WY, then AB = WY.
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Use geometric proofs to prove the following statement: if AB = CD, then 1/2AB = 1/2CD.
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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.
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Use geometric proofs to prove the following statement: RS = RS.
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Use geometric proofs to prove the following statement: if AB = RS and TU = WY, then AB + TU = RS + WY.
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Use geometric proofs to prove the following statement: if angle 1 = angle 2 and angle 2 = angle 3, then angle 1 = angle 3.
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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.
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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.
