By: James Ryden and Evan Greenberg By: James Ryden and Evan Greenberg By: James Ryden and Evan Greenberg By: James Ryden and Evan Greenberg

IIdentify and use the Algebraic Properties of Equality. IIdentify and use the Equivalence Properties of Equality and of Congruence LLink the steps of a proof by using properties and 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

 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

 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. 2.4 Building a System of Geometry Knowledge (Vocab)

AB CD.... Overlapping segments Overlapping angles

2.5 Conjectures That Lead to Theorems (Objectives) DDevelop theorems from conjectures WWrite two-column and paragraph proofs

2.5 Conjectures That Lead to Theorems (Theorems/ Postulates) VVertical Angles Theorem- If two angles form a pair of vertical angles, then they are congruent T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 T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

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

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

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

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! Congruent-Triangles-Geometry-Proof Congruent-Triangles-Geometry-Proof NOW FOR SOME FUNNY STUFF CAUSE PROOFS ARE ( not ) FUN!!!

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

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 a 2 [a 2 = ab] When with ab compared Are the same. Remove b 2. OK? [a 2 − b 2 = ab − b 2 ]  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.

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