Justification of Geometry Proofs
Justifications for Geometry Proofs – Ch 3
Properties of Equality (=): Addition: a = b → a + c = b + c Subtraction: a = b → a – c = b - c Division: a = b → a / c = b / c Multiplication: a = b → a * b = b * c Distributive: a ( b + c ) = a * b + a * c Substitution: a = b → a can be substituted for b in an equation
Properties of Congruence and Equality ( and =): Reflexive: ab = ab or Symmetric: a = b → b = a or ∠A ∠B → ∠B ∠A Transitive: a = b and b = c → a = c or ∠A ∠B and ∠B ∠C → ∠A ∠C
Definitions:
Definition of Right Angles: ∠1 is a right angle ↔ m∠1 = 90°
Definition of Perpendicular: ⟘ ↔ ∠1 is a right angle
Definition of Congruent Segments: AB = BC ↔ Definition of Congruent Angles: m∠A m∠B ↔ ∠A ∠B Definition of Midpoint: X is the midpoint of ↔ AX = XB (or ) Definition of Supplementary: ∠1 and ∠2 are supplementary ↔ m∠1 + m∠2 = 180° Definition of Complementary: ∠1 and ∠2 are complementary ↔ m∠1 + m∠2 = 90° Definition of Linear Pair: ∠1 and ∠2 are adjacent and supplementary ↔ ∠1 and ∠2 are linear pair Definition of Bisect: bisects at M ↔ CM = MD (or
Postulates:
Segment Addition Postulate: AB + BC = AC Angle Addition Postulate: m∠ABC + m∠CBD = m∠ABD Linear Pair Postulate: ∠1 and ∠2 are linear pair → ∠1 and ∠2 are supplementary
Theorems:
Vertical Angles Theorem (V.A.T. or V.A.C): ∠1 and ∠2 are vertical angles → ∠1 ∠2 Right Angle Congruence Theorem (R.A.C): ∠1 and ∠2 are right angles → ∠1 ∠2 Congruent Supplements Theorem (C.S.T.): ∠1 and ∠2 are supp and ∠1 and ∠3 are supp → ∠2 ∠3 Congruent Complements Theorem (C.C.T): ∠1 and ∠2 are comp and ∠1 and ∠3 are comp → ∠2 ∠3 Corresponding ∠ Thm (C.A.T.): if a ⃦ b, corresponding ∠s are Alternate Exterior ∠ Thm (A.E.A.T.): if a ⃦ b, alt. ext. ∠s are Alternate Interior ∠ Thm (A.I.A.T.): if a ⃦ b, alt. int. ∠s are Consecutive Interior ∠ Thm (C.I.A.T.): if a ⃦ b, consecutive int. ∠s are
Properties of Equality (=): Addition: a = b → a + c = b + c Subtraction: a = b → a – c = b - c Division: a = b → a / c = b / c Multiplication: a = b → a * b = b * c Distributive: a ( b + c ) = a * b + a * c Substitution: a = b → a can be substituted for b in an equation
Properties of Congruence and Equality ( and =): Reflexive: ab = ab or Symmetric: a = b → b = a or ∠A ∠B → ∠B ∠A Transitive: a = b and b = c → a = c or ∠A ∠B and ∠B ∠C → ∠A ∠C
Definitions:
Definition of Right Angles: ∠1 is a right angle ↔ m∠1 = 90°
Definition of Perpendicular: ⟘ ↔ ∠1 is a right angle
Definition of Congruent Segments: AB = BC ↔ Definition of Congruent Angles: m∠A m∠B ↔ ∠A ∠B Definition of Midpoint: X is the midpoint of ↔ AX = XB (or ) Definition of Supplementary: ∠1 and ∠2 are supplementary ↔ m∠1 + m∠2 = 180° Definition of Complementary: ∠1 and ∠2 are complementary ↔ m∠1 + m∠2 = 90° Definition of Linear Pair: ∠1 and ∠2 are adjacent and supplementary ↔ ∠1 and ∠2 are linear pair Definition of Bisect: bisects at M ↔ CM = MD (or
Postulates:
Segment Addition Postulate: AB + BC = AC Angle Addition Postulate: m∠ABC + m∠CBD = m∠ABD Linear Pair Postulate: ∠1 and ∠2 are linear pair → ∠1 and ∠2 are supplementary
Theorems:
Vertical Angles Theorem (V.A.T. or V.A.C): ∠1 and ∠2 are vertical angles → ∠1 ∠2 Right Angle Congruence Theorem (R.A.C): ∠1 and ∠2 are right angles → ∠1 ∠2 Congruent Supplements Theorem (C.S.T.): ∠1 and ∠2 are supp and ∠1 and ∠3 are supp → ∠2 ∠3 Congruent Complements Theorem (C.C.T): ∠1 and ∠2 are comp and ∠1 and ∠3 are comp → ∠2 ∠3 Corresponding ∠ Thm (C.A.T.): if a ⃦ b, corresponding ∠s are Alternate Exterior ∠ Thm (A.E.A.T.): if a ⃦ b, alt. ext. ∠s are Alternate Interior ∠ Thm (A.I.A.T.): if a ⃦ b, alt. int. ∠s are Consecutive Interior ∠ Thm (C.I.A.T.): if a ⃦ b, consecutive int. ∠s are