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Geometry Name Postulates, Theorems, Definitions, and Properties Used as Reasons in Proofs (through chapter 2) Postulates 1. Ruler Postulate – The points on a line can be matched one to one with real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. 2. Segment Addition Postulate – If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. ശሬሬሬሬԦ . The rays of the form ሬሬሬሬሬԦ 3. Protractor Postulate – Consider a point A on one side of ܱܤ ܱ ܣcan be matched one to one with the real numbers from 0 to 180. The measure of ∠ ܤܱܣis equal to the absolute value of the ሬሬሬሬሬԦ . difference between the real numbers for ሬሬሬሬሬԦ ܱ ܣand ܱܤ 4. Angle Addition Postulate – If P is in the interior of ∠ܴܵܶ, then ݉∠ܴܵܲ + ݉∠ܲܵܶ = ݉∠ܴܵܶ. 5. Through any two points there exists exactly one line. 6. A line contains at least two points. 7. If two lines intersect, then their intersection is exactly one point. 8. Through any three noncollinear points there exists exactly one plane. 9. A plane contains at least three noncollinear points. 10. If two points lie in a plane, then the line containing them lies in the plane. 11. If two planes intersect, then their intersection is a line. 12. Linear Pair Postulate – If two angles form a linear pair, then they are supplementary. Definitions • • • • • • • • • Congruent Segments – Segments have equal length if and only if they are congruent. Congruent Angles – Angles have equal measure if and only if they are congruent. Midpoint – A point is a midpoint if and only if it divides a segment into two congruent segments. Segment Bisector – A segment, ray, line, or plane is a segment bisector if and only if it divides a segment into two congruent segments. Angle Bisector – A ray is an angle bisector if and only if it divides an angle into two congruent angles. Complementary angles – Two angles are complementary angles if and only if the sum of their measures is 90°. Supplementary angles – Two angles are supplementary angles if and only if the sum of their measures is 180°. Perpendicular lines – Two lines are perpendicular lines if and only if they intersect to form a right angle. Right angle – An angle is a right angle if and only if it measures 90°. Algebraic Properties of Equality • • • • • • • • • • Addition Property of Equality – If a = b, then a + c = b + c Subtraction Property of Equality – If a = b, then a – c = b – c Multiplication Property of Equality – If a = b, then ac = bc Division Property of Equality – If a = b and c ≠ 0, then = Substitution Property of Equality – If a = b, then a can be substituted for b in any equation or expression. Distributive Property – ܽሺܾ + ܿሻ = ܾܽ + ܽܿ Simplify – any time you add, subtract, multiply, or divide like terms on the SAME SIDE of the equation, that is simply called “Simplify” Reflexive Property of Equality o For any real number a, a = a o For any segment ܤܣ, ܤܣ = ܤܣ. o For any angle A, ݉∠ܣ∠݉ = ܣ. Symmetric Property of Equality o If a = b, then b = a o If ܦܥ = ܤܣ, then ܤܣ = ܦܥ. o If ݉∠ܤ∠݉ = ܣ, then ݉∠ܣ∠݉ = ܤ. Transitive Property of Equality o If a = b and b = c, then a = c. o If ܦܥ = ܤܣand ܨܧ = ܦܥ, then ܨܧ = ܤܣ. o If ݉∠ ܤ∠݉ = ܣand ݉∠ܥ∠݉ = ܤ, then ݉∠ܥ∠݉ = ܣ. Theorems 2.1 Properties of Segment Congruence – Segment congruence is reflexive, symmetric, and transitive. തതതത ≅ തതതത Reflexive: For any segment ܤܣ, ܤܣ ܤܣ തതതത തതതത Symmetric: If ܦܥ ≅ ܤܣ, then ܤܣ ≅ ܦܥ. Transitive: If തതതത ܦܥ ≅ ܤܣand ܨܧ ≅ ܦܥ, then തതതത ܨܧ ≅ ܤܣ. 2.2 Properties of Angle Congruence – Angle congruence is reflexive, symmetric, and transitive. Reflexive: For any angle ܣ, ∠ܣ∠ ≅ ܣ Symmetric: If ∠ܤ∠ ≅ ܣ, then ∠ܣ∠ ≅ ܤ Transitive: If ∠ ∠ ≅ ܣand ∠ܥ∠ ≅ ܤ, then ∠ܥ∠ ≅ ܣ 2.3 Right Angle Congruence Theorem – All right angles are congruent. 2.4 Congruent Supplements Theorem – If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 2.5 Congruent Complements Theorem – If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 2.6 Vertical Angles Theorem – Vertical angles are congruent.