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Geometry: Logic

Geometry: Logic. Two Column Proofs. Do Now:. Write down your age. Multiply it by 10 Add 8 to the product Double that answer and subtract 16 Divide the result by 2 Divide the result by 5 Divide the result by 2 What happened? Did it happen to your tablemates too? Explain why this works.

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Geometry: Logic

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  1. Geometry: Logic Two Column Proofs

  2. Do Now: • Write down your age. • Multiply it by 10 • Add 8 to the product • Double that answer and subtract 16 • Divide the result by 2 • Divide the result by 5 • Divide the result by 2 What happened? Did it happen to your tablemates too? Explain why this works.

  3. Homework • Questions? • Comments? • Confusions • Ask Ask Ask!

  4. Properties of Equalities • For all these equalities, let a, b, c be any real numbers.

  5. Addition Property • If a = b, then a + c = b + c Example: 5 = 2+3, therefore 5 + 10=2 + 3 + 10 AB=BC, then AB + CD = BC + CD

  6. Subtraction Property • If a=b, then a – c = b – c Example: 5 = 2+3, therefore 5 - 10=2 + 3 - 10 AB=BC, then AB - CD = BC - CD

  7. Multiplication Property • If a=b, then a * c = b * c Example: 5 = 2+3, therefore 5 * 10= (2 + 3) * 10 AB=BC, then AB * CD = BC * CD

  8. Division Property • If a=b, then a/ c = b/ c Example: 5 = 2+3, therefore 5/10= (2 + 3)/10 AB=BC, then AB/CD = BC/CD • What’s the one condition to this property?

  9. Reflexive Property • a=a • i.e. anything is equal to itself Example: 5=5 AB=AB

  10. Symmetric Property • If a = b, then b = a Example: If 2 + 3 = 5, then 5 = 2 + 3 If AB = CD, then CD = AB

  11. Transitive Property • If a = b and b = c, then a = c Example: If 5 = 3 + 2 and 3 + 2 = 4 + 1, then 5 = 4 + 1 If AB = CD and CD = EF, then AB = EF

  12. Substitution Property • If a = b, then b can replace a in any expression. Example: 5 = 2 + 3, therefore 5*2=10 can be rewritten as (2+3)*2=10 AB= CD, therefore AB+BC=AC can be rewritten as CD+ BC = AC.

  13. Distributive Property (sum, difference) • a(b+c)= ab+ac • a(b-c)= ab-ac Example: 2( 3 + 4) = 2*3 + 2*4 AB(BC+CD)= AB*BC+ AB*CD

  14. Examples: • State the property that justifies each statement: • If 4+(-5)=-1, then x+4+ (-5)= x-1 • If 5=y, then y=5

  15. Algebraic Proof • Proof that is made up of a series of algebraic statements

  16. Example 1: • Prove that if 2x-13=-5, then x=4. Justify each step

  17. Example 2: • Prove that if -5(x+4)=70, then x= -18. Justify each step.

  18. Proof Example 3: • Given: C= 5/9(F-32) • Prove: F= 9/5F+32

  19. Example 4: • If the distance (d) moved by an object with initial velocity (u) and final velocity (v) in time (t) is given by d=t* (u+v)/2. Then u= (2d/t)-v. Prove.

  20. Prove in Geometry • Two Column Proof: contains statements and reasons organized in two columns.

  21. Example 5: • Given: <FGI is congruent to <JGK, <JGK is congruent to <KGH, m<FGH= 6x+7, and m<KGH= 8x – 5. • Prove: x=6

  22. Practice Problems • Try some on your own! • As always come find me if you are confused!

  23. Exit Ticket • Given: If (5x+1)/2 – 8 = 0 • Prove: x = 3

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