Prove Your Proof! UNIT 2
Do Now Solve the following one variable equations: ◦4m – 8 = -12 ◦X – 3.5 = 8.7 ◦4x – 7 = 8x + 3 ◦(x+8)/5 = -6 ◦2(x – 5) – 20 = 0
Have you ever developed an argument? Talk to your partner about what type of argument you would create to convince your parents/guardians to extend your curfew or increase your allowance
Objectives SWBAT form and write algebraic proofs using properties of equality and congruence SWBAT form and write geometric proofs using properties of equality and congruence
Proof An argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. Begin with a true statement and show that EACH step is valid in order to get to a valid conclusion.
Algebraic Proofs: Solving One Variable Equations Relate problem 1 in the Do Now to the properties of equality 4m – 8 = -12 4m – 8 = –12 Given equation Addition Property of Equality 4m = –4 Simplify. m =–1 Simplify. Division Property of Equality
Algebraic Proofs: Segment Addition Postulate KL + LM = KM X x – 1 = 5x – 4 3x + 2 = 5x – 4 2 = 2x – 4 6 = 2x 3 = x MLK 2x – 1X + 3 5x – 4 Segment Addition Postulate Substitution property of equality Simplify Subtraction property of equality Addition property of equality Division property of equality
Think-Pair-Compare – Write a justification for each step Given: B is the midpoint of segment AC AB = BC 1. Definition of 5y + 6 = 2y y -2y 3. 3y + 6 = y = y = 5 CBA 2y + 215y + 6
Think-Pair-Compare Solve for x. Write a justification for each step. m ABD + m DBC = m ABC 1. 3x x – 16 = 8x2. 9x – 11 = 8x 3. -9x - 9x = - x = x
Independent Practice Silently and independently complete the algebraic proofs in your guided notes
Geometric Proofs A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column.
CFU: Which properties of equality have corresponding properties of congruence? How do you decide which to use?
Two Column Proof Given: A and B are supplementary and m A = 45. Prove: m<B = 135 StatementsReasons 1. A and B are supplementary. mA = 45 Given information 2. m A + m B = 180° Definition of Supplementary m B = 180° Substitution property of equality 4. m B = 135° Subtraction property of equality
Write a justification for each step of the proof. Given: m A = 60° and m B = 2m A Prove: A and B are supplementary StatementsReasons 1. m A = 60° and m B = 2m A 2. m B = 2(60°) 3. m B = 120° 4. m A + m B = m A + m B = A and B are supplementary Given Substitution Simplify Addition Property of Equality Simplify Definition of Supplementary Angles
Check For Understanding What is the first piece of information you include in a proof? What should the last statement match up with? In a two column proof, go on the left side and are in the right. List three things you commonly use to justify steps in a proof:
Guided Practice Work with your table partner to complete the proofs in your notes Voices at a level 1 Raise your hand if you have a question
1. Given: 1 and 2 are supplementary, and 2 and 3 are supplementary. Prove: 1 3 StatementsReasons 1.Given Information 2. m 1 + m2 = 180 m2 + m3 = 180 Definition of supplementary angles 3. m 1 + m2 = m2 + m3 4. m 2 = m2 5. m 1 = m3 Subtraction property of equality 6.Definition of congruent angles
Given: ABC is a right angle. 2 3 Proof: 1 and 3 are complementary StatementsReasons 1. ABC is a right angle. 2 3 2. m ABC = 90 3.Angle Addition Postulate 4. m 1 + m 2 = 90 Substitution 5. m 2 = m 3 Definition of congruent angles 6. m 1 + m 3 = 90 7.Definition of complementary angles
Let’s wrap this up! In your own words, define a proof and the process for writing a proof (hint: use Deductive Reasoning in your response!)
Let’s wrap this up! Proof: An argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. Process: