1. 3-5 Justifying Conclusions The basis of mathematical reasoning is the making of conclusions justified by definitions, postulates, or theorems. The basis of mathematical reasoning is the making of conclusions justified by definitions, postulates, or theorems.

2. 3-5 Justifying Conclusions Example of Mathematical Proof Given:5x + 16 = 46 Prove:x = 6 Conclusions| Justifications Conclusions| Justifications 1. 5x + 16 = 46| 1. Given 1. 5x + 16 = 46| 1. Given 2. 5x = 30| 2. Addition Prop of = 2. 5x = 30| 2. Addition Prop of = 3. x = 6| 3. Mult. Prop of = 3. x = 6| 3. Mult. Prop of =

3. 3-5 Justifying Conclusions A proof of a conditional is a sequence of justified conclusions starting with the antecedent and ending with the consequent. A proof of a conditional is a sequence of justified conclusions starting with the antecedent and ending with the consequent.

4. Examples of possible justifications Point-Line Plane Postulate Point-Line Plane Postulate Line Intersection Theorem Line Intersection Theorem Definition of Parallel Lines Definition of Parallel Lines Definition of Opposite Rays Definition of Opposite Rays Distance Postulate Distance Postulate Segment Addition Postulate Segment Addition Postulate Angle Addition Postulate Angle Addition Postulate Definition of convex set Definition of convex set Law of Detachment Law of Detachment Law of Contrapositive Law of Contrapositive Definition of Midpoint Definition of Midpoint Definition of a Circle Definition of a Circle Definition of Adjacent Angles Definition of Adjacent Angles Definition of Bisector (divides into 2 equal/congruent pieces) Definition of Bisector (divides into 2 equal/congruent pieces) Definition of Complementary Definition of Complementary Definition of Supplementary Definition of Supplementary Definition of Linear Pair Definition of Linear Pair Linear Pair Theorem Linear Pair Theorem Definition of Vertical Angles Definition of Vertical Angles Vertical Angles Theorem Vertical Angles Theorem Addition Property Addition Property Multiplication Property Multiplication Property Substitution Property Substitution Property Transitive Property Transitive Property Reflexive Property Reflexive Property Symmetric Property Symmetric Property

5. 2 new justifications! Segment congruence theorem: Two segments have equal measure if and only if they are congruent. Segment congruence theorem: Two segments have equal measure if and only if they are congruent. Angle congruence theorem: Two angles have equal measure if and only if they are congruent. Angle congruence theorem: Two angles have equal measure if and only if they are congruent.

6. 3-5 Justifying Conclusions Example of Geometric Proof Given:P and Q are points on – O Prove:OP = OQ Conclusions| Justifications Conclusions| Justifications 1. P and Q are points on – O | 1. Given 1. P and Q are points on – O | 1. Given 2. OP = OQ | 2. Def. of Circle 2. OP = OQ | 2. Def. of Circle

7. 3-5 Justifying Conclusions Example of Structure of Proof Given:Where you start your proof Prove:Where you end up Conclusions| Justifications Conclusions| Justifications 1. Where you start | 1. Given 1. Where you start | 1. Given 2. Intermediate Step(s) | 2. Reason(s) 2. Intermediate Step(s) | 2. Reason(s) 3. Intermediate Step(s) | 3. Reason(s) 3. Intermediate Step(s) | 3. Reason(s) 4. Where you end up | 4. Reason 4. Where you end up | 4. Reason Justifications can be GIVENS, DEFINITIONS, POSTULATES, THEOREMS, THINGS ALEADY PROVEN IN THE PROOF.

8. 3-5 Justifying Conclusions Given:Figure at right Prove:m  1 + m  2 = 180 Conclusions| Justifications Conclusions| Justifications 1.  1 and  2 forma| 1.  1 and  2 forma| linear pair | 1. linear pair | 1. 2.  1 and  2 are| 2.  1 and  2 are| supplementary  ’s| 2. supplementary  ’s| 2. 3. m  1 + m  2 = 180| 3. 3. m  1 + m  2 = 180| 3. 1 2 n

9. 3-5 Justifying Conclusions Given:X and Y are points on – Z Prove:XZ = YZ Conclusions| Justifications Conclusions| Justifications 1. | 1. 1. | 1. 2. | 2. 2. | 2.

10. 3-5 Justifying Conclusions Given: m  1 + m  2 = 90; m  2 + m  3 = 90 Prove: m  1 = m  3 Conclusions| Justifications Conclusions| Justifications 1. | 1. 1. | 1. 2. | 2. 2. | 2. 3.| 3. 3.| 3. 1 2 3