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. 4/24/2017 2.4: Building a System of Geomtric Knowledge

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. 4/24/2017 2.4: Building a System of Geomtric Knowledge

Algebraic Properties of Equality Substitution Property of Equality: If a=b, then you may replace a with b in any equation containing a. 4/24/2017 2.4: Building a System of Geomtric Knowledge

2.4: Building a System of Geomtric Knowledge Theorems Statements that are proven true using postulates, definitions and previously proven theorems. 4/24/2017 2.4: Building a System of Geomtric Knowledge

2.4: Building a System of Geomtric Knowledge Types of Proofs 1. Two column 2. Paragraph 4/24/2017 2.4: Building a System of Geomtric Knowledge

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. 4/24/2017 2.4: Building a System of Geomtric Knowledge

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 = 12 1. 2. x – 8 + 8 = 12 + 8 2. 3. x + 0 = 20 3. 4. x = 20 4. 4/24/2017 2.4: Building a System of Geomtric Knowledge

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. 4/24/2017 2.4: Building a System of Geomtric Knowledge

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 4/24/2017 2.4: Building a System of Geomtric Knowledge

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 4/24/2017 2.4: Building a System of Geomtric Knowledge

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 4/24/2017 2.4: Building a System of Geomtric Knowledge

Let’s prove part 1 together. 4/24/2017 2.4: Building a System of Geomtric Knowledge

2.4: Building a System of Geomtric Knowledge 4/24/2017 2.4: Building a System of Geomtric Knowledge

Working with your group, prove part 2. 4/24/2017 2.4: Building a System of Geomtric Knowledge

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 4/24/2017

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 _______. 4/24/2017 2.4: Building a System of Geomtric Knowledge

Prove the Symmetric Property of Congruence for Segments. Given: AB ≅ CD Prove: CD ≅ AB 4/24/2017 2.4: Building a System of Geomtric Knowledge

2.4: Building a System of Geomtric Knowledge 4/24/2017 2.4: Building a System of Geomtric Knowledge

2.4: Building a System of Geomtric Knowledge Assignment pages 112 – 115, # 10, 12, 13 – 33 (all). 4/24/2017 2.4: Building a System of Geomtric Knowledge