Unit 3 Section 3 Logic: Intro to Proofs

If we want to prove a conditional statement is true, we start with the __________ and step-by-step, we show that the ___________ is true. At each step we have to _________________. The justification is something that is true, like h H antecedent consequent justify our conclusion Properties Definitions Postulates Theorems

Example 1 If M is the midpoint, then XM TM definition of the midpoint Given: M is the midpoint of XT. Prove: XM TM Think of this as a conditional statement, with the given as the antecedent and what you want to prove as the consequent: ________________________________. Where have you seen this before? This is one direction of the _____________________. M X If M is the midpoint, then XM TM definition of the midpoint

Example 1 Given Definition of a midpoint Given: M is the midpoint of XT. Prove: XM TM So a proof of this might look like this: M X Statements Justification M is the midpoint of XT. XM TM Given Definition of a midpoint

Example 2 Given m CAT = 60° Definition of an acute angle CAT is acute Prove: CAT is acute A T Statements Justification Given m CAT = 60° Definition of an acute angle CAT is acute

Example 3 Given Given Definition of a straight angle Now let’s go further: Given: m TAX = 120° Prove: CAT is acute C A T X Statements Justification m TAX = 120° CAX is a straight angle m CAX = 180° m CAX = m TAX + m CAT 180° = 120° + m CAT 60° = m CAT CAT is acute Given Given Definition of a straight angle Angle Addition Postulate Substitution Property Addition Property of Equality Definition of an acute angle

proofs antecedent is true vertical angle pair You were already doing problems like finding the measure of CAT, without realizing how many steps you were thinking through as you did them! The trick to doing _________ is to break down the thinking and to make sure that each step is a logical conclusion of things you already have written down. Now let’s prove our first theorem – something we have already observed in angle problems we’ve done. The Vertical Angles Congruence Theorem says “If two angles are vertical, then they are congruent.” To prove this, we assume that the _______________. That is we have a _________________. We try to prove they are congruent. proofs antecedent is true vertical angle pair

Example 4 Given Linear Pairs are supplementary Given: 2 and 4 are vertical angles Prove: 2 4 1 2 4 3 Statements Justification 1 + 2 = 180° 2 & 4 are vertical angles Given Linear Pairs are supplementary 1 + 4 = 180° Linear Pairs are supplementary 1 + 2 = 1 + 4 Substitution Property 2 = 4 Addition Property of Equality 2 4 Definition of Congruence