1. 2.4: Building a System of Geometric Knowledge
Expectations:
L3.3.1: Know the basic structure for the proof of an “if…, then” statement and that proving the contrapositive is equivalent.
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2.4: Building a System of Geomtric Knowledge

2. Algebraic Properties of Equality
Addition Property of Equality: For all real numbers a, b and c, if a = b, then a + c = b + c. Multiplication Property of Equality: For all real numbers a, b and c, if a = b, then ac = bc.
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2.4: Building a System of Geomtric Knowledge

3. Algebraic Properties of Equality
Substitution Property of Equality: If a=b, then you may replace a with b in any equation containing a.
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2.4: Building a System of Geomtric Knowledge

4. 2.4: Building a System of Geomtric Knowledge
Theorems
Statements that are proven true using postulates, definitions and previously proven theorems.
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2.4: Building a System of Geomtric Knowledge

5. 2.4: Building a System of Geomtric Knowledge
Types of Proofs
1. Two column 2. Paragraph
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2.4: Building a System of Geomtric Knowledge

6. 2.4: Building a System of Geomtric Knowledge
Parts of a Proof
1. Given: the hypothesis of the conditional. 2. Prove: the conclusion of the conditional. 3. The proof: a logical chain of statements starting with the given and ending with the prove. Each statement must be justified with a mathematical statement.
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2.4: Building a System of Geomtric Knowledge

7. 2.4: Building a System of Geomtric Knowledge
Prove: If x – 8 = 12, then x = 20
Given: x – 8 = 12 Prove: x = 20 Proof: 1. x – 8 = x – = x + 0 = x = 20 4.
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2.4: Building a System of Geomtric Knowledge

8. 2.4: Building a System of Geomtric Knowledge
Prove: If x – 8 = 12, then x = 20
Given: x – 8 = 12 Prove: x = 20 We are given x – 8 = 12 so we can use the ______________________ to add 8 to both sides. When we ________ with algebra, we get x + 0 = 20. The ________________ property tells us x = 20.
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2.4: Building a System of Geomtric Knowledge

9. Equivalence Properties of Equality
Reflexive Property of Equality: For all real numbers a, _______. Symmetric Property of Equality: For all real numbers a and b, if a = b, then ________. Transitive Property of Equality: For all real numbers a, b and c, if a = b and b = c, then ________.
Start here 9/28/04
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2.4: Building a System of Geomtric Knowledge

10. 2.4: Building a System of Geomtric Knowledge
In a proof of the vertical angle theorem, which part is assumed to be true?
two angles are congruent
two angles are vertical angles
two angles are congruent and vertical
two angles are adjacent
nothing is to be assumed true in Geometry
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2.4: Building a System of Geomtric Knowledge

11. Overlapping Segments Theorem
If A, B, C, and D are collinear points such that B is between A and C and C is between B and D such that: 1. If AB = CD, then _________. 2. If AC = BD, then _________. A D B C
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2.4: Building a System of Geomtric Knowledge

12. Let’s prove part 1 together.
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2.4: Building a System of Geomtric Knowledge

13. 2.4: Building a System of Geomtric Knowledge
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2.4: Building a System of Geomtric Knowledge

14. Working with your group, prove part 2.
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2.4: Building a System of Geomtric Knowledge

15. Overlapping Angles Theorem
1. If m∠AOB = m∠COD, then 2. If m∠AOC = m∠BOD, then
Start here 9/30/04 D C B A O
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16. Equivalence Properties for Congruence
Reflexive Property of Congruence: For all figures F, _______. Symmetric Property of Congruence: For all figures F and G, if F ≅ G, then ________. Transitive Property of Congruence: For all figures F, G and H, if F ≅ G and G ≅ H, then _______.
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2.4: Building a System of Geomtric Knowledge

17. Prove the Symmetric Property of Congruence for Segments.
Given: AB ≅ CD Prove: CD ≅ AB
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2.4: Building a System of Geomtric Knowledge

18. 2.4: Building a System of Geomtric Knowledge
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2.4: Building a System of Geomtric Knowledge

19. 2.4: Building a System of Geomtric Knowledge
Assignment
pages 112 – 115, # 10, 12, 13 – 33 (all).
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2.4: Building a System of Geomtric Knowledge