Podcast
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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Flashcards
Direct Proof
Direct Proof
A method of proof that assumes a premise is true and uses known rules to show a conclusion is also true.
Indirect Proof
Indirect Proof
A proof method that assumes the conclusion is false and shows this leads to a contradiction, proving the conclusion must be true.
Midpoint
Midpoint
A point that divides a line segment into two equal segments.
Congruent Segments
Congruent Segments
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Reflexive Property
Reflexive Property
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Symmetric Property
Symmetric Property
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Transitive Property
Transitive Property
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Substitution Property
Substitution Property
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Addition Property
Addition Property
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Subtraction Property
Subtraction Property
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Multiplication Property
Multiplication Property
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Division Property
Division Property
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Paragraph Proof
Paragraph Proof
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Two-column Proof
Two-column Proof
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Flowchart Proof
Flowchart Proof
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Simplification
Simplification
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Given
Given
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Prove
Prove
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Statement (proof)
Statement (proof)
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Reason (proof)
Reason (proof)
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Property of equality
Property of equality
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Study Notes
Direct Proof
- A direct proof assumes a premise (p) is true, then uses properties, postulates, definitions, and theorems to show that the conclusion (q) is also true.
- Steps to write a direct proof:
- State the given information.
- State what needs to be proven.
- Draw a figure (if applicable).
- Present the proof (paragraph, two-column, or flow chart).
Indirect Proof
- An indirect proof assumes the conclusion is false, then uses properties, postulates, definitions, and theorems to show that the premise is also false, leading to a contradiction.
- Steps to write an indirect proof:
- Accept the given statement as true.
- Assume the opposite of the statement to be proved.
- State the reason directly until there is a contradiction of the given or the assumed statement.
- State the assumption is false.
- Draw a figure (if applicable).
- Present the proof (paragraph, two-column, or flow chart).
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 a * c = b * c.
- Division Property: If a = b and c ≠ 0, then a/c = b/c.
- Substitution Property: If a = b, then b can be substituted for a in any expression or equation.
Example Problems/Proofs:
- Provided example problems illustrate how to apply direct and indirect proofs in geometry, including a proof that the midpoint of a line segment divides it into two congruent segments.
Activity Questions
- Questions are designed to test understanding of the different properties of equality.
Answer Key
- Correct answers to the activity questions are presented.
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Description
Explore the techniques of direct and indirect proofs in mathematics. This quiz covers the steps to write each type of proof and the properties of equality. Test your understanding of these fundamental concepts for logical reasoning.
