Algebraic and Geometric Proofs Chapter 2 Algebraic and Geometric Proofs

Section 2.1 – Counterexamples and Algebraic Proofs Learning Targets: I can make an educated guess based on inductive reasoning. I can find counterexamples. I can use algebra to write two-column proofs.

An educated guess

An example that shows that your conjecture is not true Sample: 2 * 3 = 6… Not odd. TRUE 

logical argument supported true

20 = 20 a = a If x = 20, then 20 = x. If a = b, then b = a. If x = y and y = 3, then x = 3. If a = b and b = c, then a = c. If x = 5, then x + 2 = 5 + 2 If a = b, then a + c = b + c If x = 5, then x – 2 = 5 – 2 If a = b, then a – c = b – c

If a = b, then ac = bc If x = 5, then 2x = 2(5) If a = b, then a/c = b/c (c cannot equal 0!!!) If x = 5, then x/2 = 5/2 If a = b, then a can be replaced by b in ANY equation! If -6 + 6 = 0, then 3x – 6 + 6 = 3x + 0 VERY IMPORTANT!!! a(b + c) = ab + ac 4(x + 3) = 4x + 12

statements reasons Each step of the proof (conjectures) Properties that justify each step (definitions, theorems, postulates)

3x + 5 = 17 Given 3x + 5 – 5 = 17 – 5 Subtraction Property 3x = 12 Substitution Property 3x/3 = 12/3 Division Property x = 4 Substitution Property

6x – 3 = 4x + 1 Given 6x – 3 + 3 = 4x + 1 + 3 Addition Property 6x = 4x + 4 Substitution Property 6x – 4x = 4x – 4x + 4 Subtraction Property 2x = 4 Substitution Property 2x/2 = 4/2 Division Property x = 2 Substitution Property

Homework Assignment: 2.1 Worksheet  Start the chapter on a good note! What is your goal for this chapter?

Warm Up: Conjectures and Counterexamples Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. Conjecture True or False (circle one) Counterexample if false 1. Teenagers are not good drivers. T F 2. Teachers never show movies. 3. The sum of two odd integers (Like 3 + 5) is even. 4. The product of an odd integer and an even integer (Like 5x2) is even. 5. The opposite of an integer is a negative integer. Quiz question

2.2 – Geometric Proof with Congruence Learning Targets: I can write proofs involving segment congruence. I can write proofs involving angle congruence.

AB = AB If AB = CD, then CD = AB If AB = CD and CD = EF, then AB = EF

Given Given Transitive Property

halfway

B is the midpoint of AC Given Given C is the midpoint of BD Midpoint Theorem Midpoint Theorem Transitive Property

cuts 2 equal pieces congruent

Warm Up Please complete your ACT questions in Wednesday’s section of your warm up sheet!

Warm Up

2.3 Geometric Proofs with Addition Learning Targets I can write proofs involving segment addition. I can write proofs involving angle addition.

Given Given Segment Addition Postulate (SAP) Substitution Property

straight line 180 right 90

Given Angle Addition Post. (AAP) Angle Addition Post. (AAP) Substitution Def. of congruent angles

Warm Up

Proof review Work in stations Use flip books only, no notes Write “reasons” only on your answer sheet If you have extra time at a station, start working on the problems on the back of your answer sheet

Given Given Def of Angle Bisector Def of Angle Bisector Transitive

Given Addition Property Substitution Division Property Substitution

Given Given Segment addition Segment addition Substitution Subtraction Substitution

Given Angle addition Angle addition Substitution Subtraction Substitution