Geometry Properties for Proofs

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?

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

What does the Reflexive Property of Equality state?

If a=a Signup and view all the answers

What does the Symmetric Property of Equality state?

If a=b, then b=a Signup and view all the answers

What is the Transitive Property of Equality?

If a=b, and b=c, then a=c Signup and view all the answers

What does the Angle Addition Postulate state?

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

What is the formula for calculating Area?

l × w Signup and view all the answers

What does the Distance Formula calculate?

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

What does the Midpoint Formula state?

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

What does the Triangle Sum Theorem state?

m∠A + m∠B + m∠C = 180° Signup and view all the answers

What is the formula for Circumference?

πD Signup and view all the answers

What does the Conditional state?

If p, then q Signup and view all the answers

What does the Converse state?

If q, then p Signup and view all the answers

What does the Inverse state?

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

What does the Contrapositive state?

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

What does the Distributive Property of Equality state?

a(b+c) = ab + ac Signup and view all the answers

What is the Right Angles Congruence Theorem?

All right angles are congruent Signup and view all the answers

What does the Congruent Supplements Theorem state?

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

What does the Linear Pair Postulate state?

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

What does the Reflexive Property of Congruence state?

∠A ≅ ∠A Signup and view all the answers

What does the Symmetric Property of Congruence state?

If ∠A ≅ ∠B, then ∠B ≅ ∠A Signup and view all the answers

What does the Transitive Property of Congruence state?

If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C Signup and view all the answers

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

If line AB ≅ line UV, if ∠R ≅ ∠U, and if line RT ≅ line UW, then ∆RST ≅ ∆UVW 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.