2. Powerpoint Project 5 Adam and Nikki Birnbrey

3. Properties of Equality to Know and Love  Addition 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 doesn’t = 0 Then a/c= b/c  Substitution Property- If a=b you may replace a with b in any equation containing a in the result

4. Equivalence Properties of Equality to Know and Love  Reflexive Property- For any real number a, a=a  Symmetric Property- For all real numbers a and b, if a=b, then b=a  Transitive Property- For all real numbers a, b, and c, if a=b and b=c, then a=c

5. Equivalence Relation To Know And Love = any relation that satisfies these equivalence properties  Reflexive Property- Figure A=Figure A A  Symmetric Property- If Figure A=Figure B, then Figure B=Figure A A  Transitive Property- If Figure A=Figure B and Figure B=Figure C, Then Figure A=Figure C A B C

6. Overlapping Theorem to Know and Love  Overlapping Segments- Given a segment with points A, B, C, & D arranged as shown the following statements are true: 1. If AB=CD then AC=BD 2. If AC=BD then AB=CD  Overlapping Angles- Given Angle AED with points B and C in its interior as shown, the following statements are true: 1. If Angle AEB=Angle CED then ? 2. If Angle AEC=Angle BED then ? A B C D AB E C D

7. Vertical Angles Theorem to Know and Love IIf two angles form a pair of vertical angles, then they are congruent. Given: angle 1 and 2 are vertical angles Prove: Angle 1 and 2 are congruent Statements Reasons 1.Angle 1 and 2 are vertical angles 2. Angle 1 + angle 2= 180 Angle 2 + angle 3= 180 3.Angle 1+ angle 3= angle 2 + angle 3 4.Angle 1= angle 2 (1=2) Given Linear pair property Substitution property of equality Subtraction property of equality 12 3 4

8. Congruence Supplements Theorem to Know and Love  If two angles are supplements of congruent angles then the two angles are congruent Given: Angle 1=Angle 3, Angle 1 and angle 2 are supplementary, Angle 3 and 4 are supplementary Prove: Angle 2=Angle 4 Statements Reasons 1. <1+<2=180 <3+<4=180 2. <1+<2=<3+<4 3. <1=<3 4. <1+<2=<1+<4 5. <2=<4 Definition of Supplementary Angles Transitive Given Substitution Property Subtraction Property 3412

9. More Theorems to Know and Love  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 point of intersection through twice the measure of the angle between the lines

10. Vocabulary to Know and Love  Equivalence relation= Any relation that satisfies the Reflexive, Symmetric, and Transitive Properties  Inductive reasoning= The process of forming conjectures that are based on observations  Paragraph proof= A form of a proof in which one’s reasoning is explained in paragraph form, as opposed to a two-column proof  Theorem= A statement that has been proved deductively  Two column proof= A proof in which the statements are written in the left-hand column and the reasons are given in the right-hand column  Vertical angles= The opposite angles formed by two intersecting lines

11. Practice to Know and Love (and a pooton of it) VVertical Angles: Definition, illustrated examples, and an interactive practice quiz

12. A real pooton of Practice to Know and Love (yeah math )  Given: 15x-5=10x+15 (use properties to prove)  Prove the Overlapping Segments Theorem Given: WX=YZ; Prove WY=XZ W XY Z WX=YZ WX+XY= YZ+XY WX=XY=WY segment additon postulate ______property

13. ….Extra poo  Identify the properties of equality that justify the conclusion. <B= <C; <C= <D  <B= <D ________ AB=CD; CD=AB_______ AB+BC= BC+ CD; AC=BD_______

14. Finding Values: Problemas (To Know and Love) Find x. 3X- 60 X+40 120-6X 10X