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Questions and Answers
Which property of equality states that a quantity is always equal to itself?
Which property of equality states that a quantity is always equal to itself?
If x = 5, which property of equality justifies that 5 = x?
If x = 5, which property of equality justifies that 5 = x?
If a = b and b = c, what property of equality allows us to conclude that a = c?
If a = b and b = c, what property of equality allows us to conclude that a = c?
If x = 7, what property of equality justifies replacing x with 7 in the equation x + y = 12?
If x = 7, what property of equality justifies replacing x with 7 in the equation x + y = 12?
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Which property of equality is used when we conclude that a + c = b + c from a = b?
Which property of equality is used when we conclude that a + c = b + c from a = b?
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Given m = n and c = 2, which property of equality justifies that mc = nc?
Given m = n and c = 2, which property of equality justifies that mc = nc?
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What property of equality allows you to conclude that a - d = b - d if a = b?
What property of equality allows you to conclude that a - d = b - d if a = b?
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Which property of equality allows you to divide both sides of 6p = 18 by 6 to get p = 3?
Which property of equality allows you to divide both sides of 6p = 18 by 6 to get p = 3?
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If x = y, which property of equality allows you to substitute y for x in the equation x + z = 10?
If x = y, which property of equality allows you to substitute y for x in the equation x + z = 10?
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Which property of equality justifies that 4 = 4?
Which property of equality justifies that 4 = 4?
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In a direct proof, you assume the ______ is true and use properties, postulates, definitions, and theorems to show the ______ is true.
In a direct proof, you assume the ______ is true and use properties, postulates, definitions, and theorems to show the ______ is true.
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If $a = b$ and $b = c$, what property of equality allows us to conclude that $a = c$ ?
If $a = b$ and $b = c$, what property of equality allows us to conclude that $a = c$ ?
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If $x = 7$, what property of equality justifies replacing $x$ with 7 in the equation $x + y = 12$?
If $x = 7$, what property of equality justifies replacing $x$ with 7 in the equation $x + y = 12$?
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Which property of equality is used when we conclude that $a + c = b + c$ from $a=b$?
Which property of equality is used when we conclude that $a + c = b + c$ from $a=b$?
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Given $m = n$ and $c = 2$, which property of equality justifies that $mc = n \cdot c$?
Given $m = n$ and $c = 2$, which property of equality justifies that $mc = n \cdot c$?
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If $x = y$, which property of equality allows you to substitute $y$ for $x$ in the equation $x + z = 10$?
If $x = y$, which property of equality allows you to substitute $y$ for $x$ in the equation $x + z = 10$?
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Study Notes
Direct Proof
- A direct proof starts with a premise (p) and aims to prove a conclusion (q) is true.
- It assumes p is true and uses postulates, definitions, and theorems of geometry to show q is true.
- The process typically involves steps like stating the given, stating what needs to be proven, drawing a diagram as a guide, and presenting the proof in a structured way (paragraph, two-column, or flow chart).
Writing a Direct Proof
- State the given: The given information is considered a fact.
- State what to prove: This is the desired conclusion.
- Draw a figure: This serves as a visual aid.
- Present the proof: Use a preferred method (paragraph form, two-column form, flow chart).
Indirect Proof
- Begins by assuming the conclusion (q) is false.
- Then, using the same properties, postulates, definitions, and theorems, show that the premise (p) is also false.
- This leads to a contradiction, confirming the original conclusion was correct.
Writing an Indirect Proof
- Accept the given statement is true: Start with the given information.
- Assume the opposite of the statement to be proved: Assume the conclusion is false.
- State the reason directly until there is a contradiction: Use logic and known facts to reach a contradiction
- State that the assumption of the opposite statement to be proved must be false: This confirms the original conclusion.
- Draw a figure: (Optional) for visual clarity.
- Present the proof: Use a chosen method.
Properties of Equality
- Reflexive Property: Any quantity is equal to itself (a = a)
- Symmetric Property: If a = b, then b = a.
- Transitive Property: If a = b and b = c, then a = c.
- Addition Property: If a = b, then a + c = b + c.
- Subtraction Property: If a = b, then a – c = b – c.
- Multiplication Property: If a = b, then ac = bc.
- Division Property: If a = b and c ≠ 0, then a/c = b/c.
- Substitution Property: If a = b, then b can replace a in any expression or equation.
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Description
This quiz covers the concepts of direct and indirect proofs in geometry. It outlines the steps for writing both types of proofs, emphasizing the importance of premises, conclusions, and visual aids. Test your understanding of how to structure and present geometric proofs effectively.
