Geometry Properties for Proofs - Flashcards
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Questions and Answers

What does the Addition Property of Equality state?

  • If a=b, then a-c=b-c
  • If a=b, then ac=bc
  • If a=b, then a+c=b+c (correct)
  • If a=b, and c≠0, then a/c=b/c

    • What is the definition of the Subtraction Property of Equality?

  • If a=b, then ac=bc
  • If a=b, then a+c=b+c
  • If a=b, then a-c=b-c (correct)
  • If a=b, and c≠0, then a/c=b/c

    • What does the Multiplication Property of Equality state?

  • If a=b, then ac=bc (correct)
  • If a=b, then a+c=b+c
  • If a=b, then a-c=b-c
  • If a=b, and c≠0, then a/c=b/c

    • What is the definition of the Division Property of Equality?

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

    What does the Reflexive Property of Equality state?

    <p>If a=a</p> Signup and view all the answers

    What does the Symmetric Property of Equality state?

    <p>If a=b, then b=a</p> Signup and view all the answers

    What is the Transitive Property of Equality?

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

    What does the Angle Addition Postulate state?

    <p>If P is in the interior of ∠RST, then m∠RST = m∠RSP + m∠PST</p> Signup and view all the answers

    What is the formula for calculating Area?

    <p>l × w</p> Signup and view all the answers

    What does the Distance Formula calculate?

    <p>AB = √((x₂-x₁)² + (y₂-y₁)²)</p> Signup and view all the answers

    What does the Midpoint Formula state?

    <p>{(x₁+x₂)/2}, {(y₁+y₂)/2}</p> Signup and view all the answers

    What does the Triangle Sum Theorem state?

    <p>m∠A + m∠B + m∠C = 180°</p> Signup and view all the answers

    What is the formula for Circumference?

    <p>πD</p> Signup and view all the answers

    What does the Conditional state?

    <p>If p, then q</p> Signup and view all the answers

    What does the Converse state?

    <p>If q, then p</p> Signup and view all the answers

    What does the Inverse state?

    <p>If not p, then not q</p> Signup and view all the answers

    What does the Contrapositive state?

    <p>If not q, then not p</p> Signup and view all the answers

    What does the Distributive Property of Equality state?

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

    What is the Right Angles Congruence Theorem?

    <p>All right angles are congruent</p> Signup and view all the answers

    What does the Congruent Supplements Theorem state?

    <p>If ∠1 and ∠2 are supplementary and ∠3 and ∠3 are supplementary, then ∠1 ≅ ∠3</p> Signup and view all the answers

    What does the Linear Pair Postulate state?

    <p>∠1 and ∠2 form a linear pair, so ∠1 and ∠2 are supplementary and m∠1 + m∠2 = 180°</p> Signup and view all the answers

    What does the Reflexive Property of Congruence state?

    <p>∠A ≅ ∠A</p> Signup and view all the answers

    What does the Symmetric Property of Congruence state?

    <p>If ∠A ≅ ∠B, then ∠B ≅ ∠A</p> Signup and view all the answers

    What does the Transitive Property of Congruence state?

    <p>If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C</p> Signup and view all the answers

    What does the Side-Angle-Side (SAS) Congruence Postulate state?

    <p>If line AB ≅ line UV, if ∠R ≅ ∠U, and if line RT ≅ line UW, then ∆RST ≅ ∆UVW</p> Signup and view all the answers

    Study Notes

    Properties of Equality

    • 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, a can replace b in any equation or expression.
    • Reflexive Property: a = a, indicating that any quantity is equal to itself.
    • Symmetric Property: If a = b, then b = a, showcasing equality’s two-way nature.
    • Transitive Property: If a = b and b = c, then a = c, establishing a chain of equality.

    Properties of Congruence

    • Reflexive Property of Congruence: ∠A ≅ ∠A, meaning an angle is congruent to itself.
    • Symmetric Property of Congruence: If ∠A ≅ ∠B, then ∠B ≅ ∠A, illustrating the mutual relationship between congruent angles.
    • Transitive Property of Congruence: If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.

    Geometric Postulates and Theorems

    • Segment Addition Postulate: If B is between A and C, then AB + BC = AC, highlighting how lengths combine.
    • Midpoint Formula: Given two points {(x₁, y₁) and (x₂, y₂)}, the midpoint is {(x₁+x₂)/2, (y₁+y₂)/2}.
    • Distance Formula: The distance between points A(x₁, y₁) and B(x₂, y₂) is AB = √[(x₂ - x₁)² + (y₂ - y₁)²].
    • Angle Addition Postulate: If point P is inside ∠RST, then m∠RST = m∠RSP + m∠PST.

    Area and Perimeter

    • Area: Calculated as length times width (l × w).
    • Perimeter: Found by adding all sides (l + l + w + w).
    • Circumference: The perimeter of a circle, computed using πD, where D is the diameter.

    Logical Statements in Geometry

    • Conditional Statement: In the form "if p, then q."
    • Converse: The reverse of a conditional statement, "if q, then p."
    • Inverse: Negates both parts of the conditional, "if not p, then not q."
    • Contrapositive: Reverse and negate, "if not q, then not p."

    Additional Theorems

    • Distributive Property: a(b + c) = ab + ac, allowing for distribution of multiplication over addition.
    • Right Angles Congruence Theorem: All right angles are congruent.
    • Congruent Supplements Theorem: If two angles are each supplementary to the same angle, they are congruent.
    • Congruent Complements Theorem: If two angles are complementary to the same angle, they are congruent.
    • Linear Pair Postulate: If two angles form a linear pair, they are supplementary (m∠1 + m∠2 = 180°).
    • Triangle Sum Theorem: The sum of the angles in a triangle equals 180°.

    Triangle Congruence Properties

    • Reflexive Property of Congruent Triangles: ∆ABC ≅ ∆ABC, showing that a triangle is congruent to itself.
    • Symmetric Property of Congruent Triangles: If ∆ABC ≅ ∆DEF, then ∆DEF ≅ ∆ABC.
    • Transitive Property of Congruent Triangles: If ∆ABC ≅ ∆DEF and ∆DEF ≅ ∆JKL, then ∆ABC ≅ ∆JKL.
    • Side-Angle-Side (SAS) Congruence Postulate: If two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent (∆RST ≅ ∆UVW if line AB ≅ line UV, ∠R ≅ ∠U, line RT ≅ line UW).

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    Test your knowledge on essential properties of equality used in geometry proofs with this set of flashcards. Each card covers a fundamental property, including addition, subtraction, multiplication, division, and substitution. Perfect for students looking to master geometry concepts.

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