Ch. 2 2.4- 2.5

1. Ch. 22.4- 2.5 By: James Ryden and Evan Greenberg By: James Ryden and Evan Greenberg Ch. 22.4- 2.5 By: James Ryden and Evan Greenberg By: James Ryden and Evan Greenberg

2. 2.4 Building a System of Geometry Knowledge (Objectives) • Identify and use the Algebraic Properties of Equality. • Identify and use the Equivalence Properties of Equality and of Congruence • Link the steps of a proof by using properties and postulates

3. 2.4 Building a System of Geometry Knowledge (Theorems/ Postulates) • Additio Property- if a=b, then a+c=b+c • Subtraction Property- if a=b, then a-c=b-c • Multiplication Property- if a=b, then ac=bc • Division Property- if a=b, and c does not =0, then a/c=b/c • Substitution Property- if a=b, you may replace a with b in any true equation containing a and the resulting equation will be still be true • Overlapping Segments Theorem-given a segment of points a, b, c, and d (in order) the following statements are true: • if ab=cd then ac=bd • if ac=bd then ab=cd

4. 2.4 Building a System of Geometry Knowledge (Theorems/Postulates (cont)) • Reflexive Property of Equality-for any real number a, a=a • Symmetric Property of Equality-for all real numbers a and b, if a=b then b=a • Transitive Property of Equality-for all real numbers a, b,c, if a=b and b=c, then a=c • Reflexive Property of Congruence-if figure a is congruent to figure b, then figure b is congruent to figure a • Overlapping Angles Theorem-Given <aod with points b and c in its interior, the following statements are true: • if m<aob=mcod, then m<aoc=mbod • if m<aoc=mbod, then m<aob=mcod

5. 2.4 Building a System of Geometry Knowledge (Vocab) • Equivalence Relation-Any relation that satisfies the reflexive property, symmetric property, and/or transitive property. • Paragraph Proof-An alternative to the two-column proof where one writes out a paragraph instead of two columns. • Theorem-A statement that has been proved deductively. • Two-Column Proof-A proof written out in two columns, one with statements, and the other with the reasons behind the statements.

6. 2.4 Building a System of Geometry Knowledge (Diagrams) C D A B . . . . Overlapping segments Overlapping angles

7. 2.5 Conjectures That Lead to Theorems (Objectives) • Develop theorems from conjectures • Write two-column and paragraph proofs

8. 2.5 Conjectures That Lead to Theorems (Theorems/ Postulates) • Vertical Angles Theorem- If two angles form a pair of vertical angles, then they are congruent • Theorem- Reflection across two parallel lines is equivalent to a translation of twice the distance between the lines and in a direction perpendicular to the lines • Theorem- Reflection across two intersecting lines is equivalent to a rotation about the piont of intersection through twice the measure of the angle between the line

9. 2.5 Conjectures That Lead to Theorems (Vocab) • Inductive Reasoning-is the process of forming conjectures that are based on observations • Vertical Angles-are the opposite angles formed by two intersecting lines

10. 2.5 Conjectures That Lead to Theorems (Diagrams) Vertical Angles Angles Supplementary

11. Review 1).List and define all equivalence properties of equality 2).List and define all equivalence properties of congruence 3). <1, <2, <3, <4 are vertical angles. Prove <1 is congruent to <2

12. HELPFUL WEBSITE ALERT! • This website is really really really really really good for helping you with writing proofs. I went from a 13% to an 80% by using this website! http://www.wikihow.com/Write-a-Congruent-Triangles-Geometry-Proof NOW FOR SOME FUNNY STUFF CAUSE PROOFS ARE (not) FUN!!!

13. Remember when Ms. Bradley always told us to find x…

14. And here’s something that will really get you thinking… • If a = b (so I say)                                        [a = b] And we multiply both sides by a Then we'll see that a2                                  [a2 = ab] When with ab compared Are the same. Remove b2. OK?                 [a2− b2 = ab − b2] • Both sides we will factorize. See? Now each side contains a − b.                     [(a+b)(a − b) = b(a − b)] We'll divide through by a Minus b and olé a + b = b. Oh whoopee!                                [a + b = b] • But since I said a = b b + b = b you'll agree?                                 [b + b = b] So if b = 1 Then this sum I have done                          [1 + 1 = 1] Proves that 2 = 1. Q.E.D.

15. Unusual Theorems • Theorem . A sheet of writing paper is a lazy dog. • Proof: A sheet of paper is an ink-lined plane. An inclined plane is a slope up. A slow pup is a lazy dog. Therefore, a sheet of writing paper is a lazy dog. • Theorem . A peanut butter sandwich is better than eternal happiness. • Proof: A peanut butter sandwich is better than nothing. But nothing is better than eternal happiness. Therefore, a peanut butter sandwich is better than eternal happiness.  • Theorem . Christmas = Halloween = Thanksgiving (at least for assembly language programmers). • Proof: By definition, Christmas = Dec. 25; Halloween = Act. 31; Thanksgiving = Nov. 27, sometimes. Again by definition: • Dec 25 is 25 base 10 or (2 x 10) + (5 x 1) = 25. • Oct 31 is 31 base 8 or (3 x 8) + (1 x 1) = 25. • Nov 27 is 27 base 9 or (2 x 9) + (7 x 1) = 25.13