2.6 Proving Statements about Angles

2.6 Proving Statements about Angles p. 109

Warm-up Find the measure of angles 1 – 12:

2.6 Warm-up • Find the measure of each angle • The right angle: ___________ • The complement of 42: _________ • The supplement of 42: ____________ • Two congruent angles that are complementary: ____________ • Two congruent angles that are supplementary: _______________

Review of Concepts inductive reasoning: is a bringing forward of separate facts or instances so as to prove a general statement" deductive reasoning going from a known principle to an unknown, from the general to the specific or from premise to a logical conclusion

Properties of Equality

Definitions • Given: • Information that is provided up front • You do not have to make any conclusions or apply any theorems to use it • Prove: • Facts that you are requested to prove using the information that is given and previously learned theorems and postulates

Theorem • A true statement that follows as a result of other true statements. • All theorems MUST be proven! • Postulates and axioms are used to prove theorems

Theorem 2.2 – Properties of Angle Congruence • Angle congruence is reflexive, symmetric, and transitive.

2-Column Proof • Numbered statements and corresponding reasons in a logical order organized into 2 columns. statements reasons 1. 1. 2. 2. 3. 3. etc.

All the men in a certain room are bakers, All bakers get up early to bake bread in the morning, Jim is in that specific room. Jim gets up early in the morning Given Given Given Deductive Reasoning Given: All men in a room are bakers Jim is in that room Bakers get up earlyProve: Jim gets up early in the morning

Statements <A <B; <B <C m<A=m<B; m<B=m<C m<A=m<C <A <C * You will be proving the reflexive & symmetric parts for homework. Reasons Given Defn. of <s Trans. Prop. Of = Defn. of <s Proof of transitive part:Given: <A<B and <B <CProve: <A <C

Statements <A & <B are right <s. m<A=90o; m<B=90o m<A=m<B <A <B Reasons Given Defn. of rt. < Subst. prop of = Defn. of <s Thm 2.3 – Right Angle Congruence Thm. All right <s are . <A & <B are right <s.

Thm 2.4 - supplements thm. • If 2 <s are supplementary to the same < (or to <s), then they are . • If <1 & <2 are suppl & <2 & <3 are suppl, then <1 <3. 1 2 3

Properties of Equalities with Length and Measure

Example of Angle Proof

Statements <1 & <2 are suppl.; <2 & <3 are suppl. m<1+m<2=180o; m<2+m<3=180o m<1+m<2=m<2+m<3 m<1=m<3 <1 <3 Reasons Given defn,. of suppl <s Subst. prop of = - prop of = Defn. of <s Proof of congruent suppl. Thm.

Thm. 2.5 – Congruent complements thm • If 2 <s are complementary to the same < (or to <s), then they are . • If <1 & <2 are complementary, and <2 & <3 are complementary, then <1 <3. • Proof is almost identical to the last thm., just change suppl. to compl. 1 2 3

Postulate 12 – Linear Pair Post. • If 2 <s form a linear pair, then they are supplementary. • <1 & <2 are a linear pair, therefore they are suppl. 1 2

Thm. 2.6 – Vertical <s Thm. • Vertical angles are . • <1 <3 & <2 <4. 2 4 1 3

Statements <1 & <3 are vert. <2 & <4 are vert. 2. <1 & <2 are a lin pr.; <3 & <2 are a lin. pr. 3. <1 & <2 are suppl; <3 & <2 are suppl. 4. <1 <3 Reasons Given Defn. of Lin. Pr. Ln. Pr. Post. Suppls. Thm. Proof of Vertical <s thm

Deductive reasoning is the process by which a person makes conclusions based on previously known facts. An instance of deducti a person knows that all the men in a certain room are bakers, that all bakers get up early to bake bread in the morning, and that Jim is in that specific room. Knowing these statements to be true, a person could deductively reason that Jim gets up early in the morning ve reasoning might go something like this: a person knows that all the men in a certain room are bakers, that all bakers get up early to bake bread in the morning, and that Jim is in that specific room. Knowing these statements to be true, a person could deductively reason that Jim gets up early in the morning. Such a method of reasoning is a step-by-step process of drawing conclusions based on previously known truths. Usually a general statement is made about an entire class of things, and then one specific example is given. If the example fits into the class of things previously mentioned, then deductive reasoning can be used. Deductive reasoning is the method by which conclusions are drawn in geometric proofs.

2.6 Proving Statements about Angles