Powerpoint Project 5 Adam and Nikki Birnbrey Properties of Equality to Know and Love Addition...
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Transcript of Powerpoint Project 5 Adam and Nikki Birnbrey Properties of Equality to Know and Love Addition...
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_______