Powerpoint Project 5

Adam and Nikki Birnbrey

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

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

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

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

A B

E

CD

Vertical Angles Theorem to Know and Love

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= 180Angle 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

1 2

3

4

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

3 4 1 2

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

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

Practice to Know and Love(and a pooton of it)

Vertical Angles: Definition, illustrated examples, and an interactive practice quiz

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 X Y Z

WX=YZ

WX+XY= YZ+XY

WX=XY=WY

segment additon postulate

______property

….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_______

Finding Values: Problemas (To Know and Love)

Find x.

3X- 60

X+40

120-6X

10X