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- 1. M AT H8 - Q U A R T E R 3 AXIOMS/POSTULATES THEOREMS Postulate 1 Postulate 2 Postulate 3 Postulate 4 Postulate 5 Postulate 6 Theorem 1 Theorem 2 Theorem 3 Postulate 7
- 2. M AT H8 - Q U A R T E R 3 AXIOMS /POSTULATES THEOREMS Postulate 1 Postulate 2 Postulate 3 Postulate 4 Postulate 5 Postulate 6 Theorem 1 Theorem 2 Theorem 3 Point-existence Postulate Line Postulate Plane Postulate Flat-Plane Postulate Plane-Intersection Postulate Segment Addition Postulate Postulate 7 Angle Addition Postulate
- 3. M AT H8 - Q U A R T E R 3 A line contains at least two points If points A, B and C are collinear and B is between A and C then AB + BC = AC If point G lies in the interior of angle ∠DAF then m∠DAG + m∠GAF= m∠DAF If two distinct planes intersect, then their intersection is a line SEGMENT ADDITION A N G L E A D D I T I O N PLANE INTERSEC TION POINT EXISTENCE Two points determine exactly one line LINE POSTULATE If two points of a line are in a plane, then the line is in the plane. FLAT PLANE Any three noncollinear points lie in exactly one plane. PLANE POS P O S T U L AT E S
- 4. M AT H8 - Q U A R T E R 3 AXIOMS • A statement or proposition that is being regarded or accepted as true. • Axiom is being used in other sections of mathematics. • Postulate, which is also accepted true without any proof, is usually used in geometry.
- 5. M AT H8 - Q U A R T E R 3 AXIOMS Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality If 𝒙 − 𝟓 = 𝟏𝟎 then If 𝒙 + 𝟑 = 𝟐𝟓 then If 𝟏 𝟐 𝒂 = 𝟏𝟎, then If 𝟐𝒂 = 𝟏𝟎, then Given 𝒙 = 𝒚 + 𝟏 and 𝒚 = 𝟏, then 𝒙 = 𝟐 𝒂 = 𝒂, 𝒃 = 𝒃 𝒂 = 𝒃, 𝒃 = 𝒂 𝒂 = 𝒃, and 𝒃 = 𝒄, then 𝒂 = 𝒄 𝒙 = 𝟏𝟓 𝒙 = 𝟐𝟐 𝒂 = 𝟐𝟎 𝒂 = 𝟓
- 6. M AT H8 - Q U A R T E R 3 AXIOMS Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality If 𝒂 = 𝒃 and 𝒄 = 𝒅, then 𝒂 + 𝒄 = 𝒃 + 𝒅 If 𝒂 = 𝒃 and 𝒄 = 𝒅, then 𝒂 − 𝒄 = 𝒃 − 𝒅 If 𝒂 = 𝒃, then 𝒂𝒄 = 𝒃𝒅 If 𝒂 = 𝒃 and ≠ 0 , then 𝒂 𝒄 = 𝒃 𝒅 Given 𝒙 = 𝒚 + 𝟏 and 𝒚 = 𝟏, then 𝒙 = 𝟐 𝒂 = 𝒂, 𝒃 = 𝒃 𝒂 = 𝒃, 𝒃 = 𝒂 𝒂 = 𝒃, and 𝒃 = 𝒄, then 𝒂 = 𝒄
- 7. M AT H8 - Q U A R T E R 3 AXIOMS AB AB ≅ P R O P E R T Y O F C O N G R U E N C E AB PR ≅ PR AB ≅ Reflexive Property of Congruence Symmetric Property of Congruence If ∠A ≅ ∠B and ∠B ≅ ∠C then ∠A ≅ ∠C Transitive Property of Congruence
- 8. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: State the axiom that will justify each statement. ___________________ 1. In the equation 5 + x = 10, what property should be used to solve this equation? ___________________ 2. What property of congruence is used in “if a ≅ b, then b ≅ a” ? ___________________ 3. What property should be used to solve the equation 3x = 12?
- 9. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: State the axiom that will justify each statement. ___________________ 4. What property should be used to solve the equation 𝑥 𝟒 = 15? ___________________ 5. If 3x – 5 = 7 and x = 4, then 3(4) – 5 = 7. What property is being illustrated?
- 10. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: Use the given property to complete the statement 6. Symmetric Property : If ∠ J ≅ ∠ K, then ______________ 7. Substitution Property : If AB – CD = 15 and CD = 7, then _____________________. 8. Transitive Property : If m ∠ A + m ∠ B = m ∠ C and m ∠ C = m ∠ D, then _________.
- 11. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: Use the given property to complete the statement 9. Division Property : If 2(m ∠ A) = 14, then ______________ 10. Subtraction Property : If 25x + 12 = 32, then _____________
- 12. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: State the axiom that will justify each statement. ___________________ 1. In the equation 5 + x = 10, what property should be used to solve this equation? ___________________ 2. What property of congruence is used in “if a ≅ b, then b ≅ a” ? ___________________ 3. What property should be used to solve the equation 3x = 12? Subtraction Property of Equality Symmetric Property of Congruence Division Property of Equality
- 13. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: State the axiom that will justify each statement. ___________________ 4. What property should be used to solve the equation 𝑥 𝟒 = 15? ___________________ 5. If 3x – 5 = 7 and x = 4, then 3(4) – 5 = 7. What property is being illustrated? Multiplication Property of Equality Substitution Property of Equality
- 14. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: Use the given property to complete the statement 6. Symmetric Property : If ∠ J ≅ ∠ K, then ______________ 7. Substitution Property : If AB – CD = 15 and CD = 7, then _____________________. 8. Transitive Property : If m ∠ A + m ∠ B = m ∠ C and m ∠ C = m ∠ D, then _________. ∠ K ≅ ∠ J AB - 7 = 15 m∠A + m∠B = m∠D
- 15. M AT H8 - Q U A R T E R 3 G1 Name Date Section SW # 4 Direction: Use the given property to complete the statement 9. Division Property : If 2(m ∠ A) = 14, then ______________ 10. Subtraction Property : If 25x + 12 = 32, then _____________ m∠ A = 7 𝟐𝟓𝒙 = 𝟐𝟎
- 16. THEOREMS M o du l e 1
- 17. M AT H8 - Q U A R T E R 3 MATHEMATICAL SYSTEM Undefined Terms Terms which cannot be precisely defined. Defined Terms Terms have a formal definition. Axioms/ Postulates A statement which is accepted as true without proof. Theorems A statement accepted after proven true deductively.
- 18. M AT H8 - Q U A R T E R 3 THEOREMS • a theorem is a statement that has to be proven before being accepted. • The proof in the theorem is a sequence of true facts such as undefined terms, defined terms, axioms or postulates, and even previously proven theorems that are arranged in a logical order.
- 19. M AT H8 - Q U A R T E R 3 AXIOMS/POSTU LATES THEORE MS Postulate 1 Postulate 2 Postulate 3 Postulate 4 Postulate 5 Postulate 6 Theorem 1 Theorem 2 Theorem 3 Point-existence Postulate Line Postulate Plane Postulate Flat-Plane Postulate Plane-Intersection Postulate Segment Addition Postulate Postulate 7 Angle Addition Postulate
- 20. M AT H8 - Q U A R T E R 3 Theorem 1 LINE INTERSECTION THEOREM If two lines intersect, their intersection contains only one point A m n Line m and line n intersect at point A
- 21. M AT H8 - Q U A R T E R 3 AXIOMS/POSTU LATES THEORE MS Postulate 1 Postulate 2 Postulate 3 Postulate 4 Postulate 5 Postulate 6 Theorem 1 Theorem 2 Theorem 3 Point-existence Postulate Line Postulate Plane Postulate Flat-Plane Postulate Plane-Intersection Postulate Segment Addition Postulate Postulate 7 Angle Addition Postulate Line Intersection Theorem
- 22. M AT H8 - Q U A R T E R 3 Theorem 2 LINE-PLANE INTERSECTION THEOREM If two lines intersect, their intersection contains only one point A m n Line m and line n intersect at point A
- 23. M AT H8 - Q U A R T E R 3 AXIOMS/POSTU LATES THEORE MS Postulate 1 Postulate 2 Postulate 3 Postulate 4 Postulate 5 Postulate 6 Theorem 1 Theorem 2 Theorem 3 Point-existence Postulate Line Postulate Plane Postulate Flat-Plane Postulate Plane-Intersection Postulate Segment Addition Postulate Postulate 7 Angle Addition Postulate Line Intersection Theorem Line Plane Theorem
- 24. M AT H8 - Q U A R T E R 3 Theorem 2 TRIANGLE-ANGLE- SUM THEOREM In every triangle the sum of the measures of the three interior angles is 180° m ∠ A+m ∠ B+m ∠ C=180° A B C
- 25. M AT H8 - Q U A R T E R 3 AXIOMS/POSTU LATES THEORE MS Postulate 1 Postulate 2 Postulate 3 Postulate 4 Postulate 5 Postulate 6 Theorem 1 Theorem 2 Theorem 3 Point-existence Postulate Line Postulate Plane Postulate Flat-Plane Postulate Plane-Intersection Postulate Segment Addition Postulate Postulate 7 Angle Addition Postulate Line Intersection Theorem Line Plane Theorem Triangle Angle Theorem