1. Friday, September 13th Z is the centroid. If Line 2. XZ = 5. Find XL ZK= 10 Find the median KW Warm Up

2. Quiz Answers

3. Part 1: Centroid

4. Vertex to centroid=2/3 the distance of the median Midsegment to centroid = 1/3 the distance of the median

5. Part 2: Circumcenter

6. All Vertexes to circumcenter are EQUAL All corresponding sides are equal

7. Homework Answers

8. Proofs

9. On a sheet of paper, write down how you would make a peanut butter and jelly sandwich

10. When we write proofs, we have to write every step!!!

11. Part I Algebraic Proofs

12. Subtraction Property of Equality Subtracting the same number to both sides of an equation does not change the equality of the equation. If a = b, then a – c = b – c. Ex: x = y, so x – 4 = y – 4

13. Addition Property of Equality Adding the same number to both sides of an equation does not change the equality of the equation. If a = b, then a + c = b + c. Ex: x = y, so x + 4 = y +4

14. Multiplication Property of Equality Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. • If a = b, then ac = bc. • Ex: x = y, so 3x = 3y

15. Division Property of Equality Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation. • If a = b, then a/c = b/c. • Ex: x = y, so x/7 = y/7

16. Commutative Property Changing the order of addition or multiplication does not matter. “Commutative” comes from “commute” or “move around”, so the Commutative Property is the one that refers to moving stuff around.

17. Commutative Property Addition: a + b = b + a Ex: 1 + 9 = 9 + 1

18. Commutative Property Multiplication: a ∙ b = b ∙ a Ex: 8 ∙ 6 = 6 ∙ 8

19. Associative Property The change in grouping of three or more terms/factors does not change their sum or product. “Associative” comes from “associate” or “group”, so the Associative Property is the one that refers to grouping.

20. Associative Property • Addition: a + (b + c) = (a + b) + c • Ex: 1 + (7 + 9) = (1 + 7) + 9

21. Associative Property Multiplication: a ∙ (b∙ c) = (a ∙ b) ∙ c Ex: 8 ∙ (3 ∙ 6) = (8 ∙ 3) ∙ 6

22. Distributive Property The product of a number and a sum is equal to the sum of the individual products of terms.

23. Distributive Property a ∙ (b + c) = a ∙ b + a ∙ c Ex: 5 ∙ (x + 6) = 5 ∙ x + 5∙ 6

24. Reflexive Property Think: Mirror A= A -3 = -3 M<1= m<1

25. Symmetric Property If a = b then b = a -3 = x then X= - 3 If xy is vwthen vw is xy

26. Transitive Property For any numbers a, b, and c, if a = b and b = c, then a = c If < 1 <2, and <2 <3 then <1 <3

27. Remember! The Distributive Property states that a(b + c) = ab + ac.

28. Division Property of Equality Solve the equation 4m – 8 = –12. Write a justification for each step. 4m – 8 = –12 Given equation +8+8Addition Property of Equality 4m = –4 Simplify. m = –1 Simplify.

29. Solve the equation . Write a justification for each step. Given equation Multiplication Property of Equality. t = –14 Simplify.

30. Identifying Property of Equality and Congruence Identify the property that justifies each statement. A. QRS  QRS B. m1 = m2 so m2 = m1 C. AB  CD and CD  EF, so AB EF. D. 32° = 32° Reflex. Prop. of . Symm. Prop. of = Trans. Prop of  Reflex. Prop. of =

31. Given z – 5 = –12 Mult. Prop. of = z = –7 Add. Prop. of = Review Solve each equation. Write a justification for each step. 1.

32. Given 6r – 3 = –2(r + 1) 6r – 3 = –2r – 2 Distrib. Prop. Add. Prop. of = 8r – 3 = –2 8r = 1 Add. Prop. of = Div. Prop. of = Solve each equation. Write a justification for each step. 2.6r – 3 = –2(r + 1)

33. Identify the property that justifies each statement. 3. x = y and y = z, so x = z. 4. DEF  DEF 5. ABCD, so CDAB. Trans. Prop. of = Reflex. Prop. of  Sym. Prop. of 