2.4: building a system of geometric knowledge expectations: l3.3.1: know the basic structure for the...
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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.
04/20/232.4: Building a System of Geomtric
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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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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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Theorems
Statements that are proven true using postulates, definitions and previously proven theorems.
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Types of Proofs
1. Two column
2. Paragraph
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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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Prove: If x – 8 = 12, then x = 20
Given: x – 8 = 12Prove: x = 20Proof:1. x – 8 = 12 1. 2. x – 8 + 8 = 12 + 8 2. 3. x + 0 = 20 3. 4. x = 20 4.
04/20/232.4: Building a System of Geomtric
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Prove: If x – 8 = 12, then x = 20
Given: x – 8 = 12 Prove: x = 20We 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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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 ________.
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In a proof of the vertical angle theorem, which part is assumed to be true?
A.two angles are congruentB.two angles are vertical anglesC.two angles are congruent and verticalD.two angles are adjacentE.nothing is to be assumed true in Geometry
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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 _________.
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A B C D
Let’s prove part 1 together.
04/20/232.4: Building a System of Geomtric
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04/20/232.4: Building a System of Geomtric
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Working with your group, prove part 2.
04/20/232.4: Building a System of Geomtric
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Overlapping Angles Theorem
1. If m AOB = m COD, then ∠ ∠
2. If m AOC = m BOD, then ∠ ∠
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DCBA
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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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Prove the Symmetric Property of Congruence for Segments.
Given: AB ≅ CDProve: CD ≅ AB
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04/20/232.4: Building a System of Geomtric
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Assignment
pages 112 – 115, # 10, 12, 13 – 33 (all).
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