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Lesson2-2

Page history last edited by Math in a Box - Susan Johnsey gm 1 year, 1 month ago

 

Lesson 2-2

   Properties from Algebra that we use in GEOMETRY

        by Susan Johnsey

Click the Wiki tab above to return to the Lessons Listing, or

go to www.mathinabox.com/Classes.html for info on all my classes.

 

The BIG GREEN box below is very important for GEOMETRY PROOFS.

Learn it well  !!  Write it all down!   Will you?

  

  

 We will study the lesson over 2 days.

DAY  1   

 

You really need to read my comments below.     Then read the book.

 

Add the eleven properties  to your Geometry Collection .          

The first day you need to study the properties on page 37.   

 

The first 4 properties (add , subtract, multiply and divide) are used often by students to solve equations.


 

 ADDITION Property  of Equality 

      add equal amounts to both sides of equation.

 

 If we have x - 2 = -6  then we can solve by adding 2 to both sides of the equation.

      left side = right side

     x - 2   =  -6          We use the ADDITION Property .   We add 2 on BOTH sides.

    x - 2 +2  = -6 +2.         On LEFT side of = I added 2 and on right side I added 2.

          x= - 4.  

 

WARNING:

The ADDITION PROPERTY  is used when

    we add EQUAL amounts to BOTH sides of an EQUATION.    

ADDing the same thing on both sides of equation is the ADDITION property.

 left side= right side

     3x+5 =  

  3x+5+5 = +5

can become  3x+10=  14.

Do you see that  I added 5 to BOTH sides? That is the Addition Property.

 

 The   ADDITION PROPERTY  is NOT when we add two terms or two numbers or two whatever.  It is NOT   7+8=15 or 6x+x= 7x or segment AB + segment BC.   It is adding to both SIDES of an EQUATION.  

 

And for GEOMETRY:

        angle 4 + 30 degrees = 180 degrees.   We could subtract 30 degrees from both sides.   

        We would have angle 4 = 150 degrees.     That is the SUBTRACTION property.

 

And for GEOMETRY:

  Angle M is a right angle thus

              angle M = 90 by the definition of right angle.   

   We can divide both sides by 2 and know that

              1/2 angle M = 45 degrees.   

           That is the DIVISION  property.

 

 

BUT what property are we using when we REPLACE

80 degrees+70 degrees with 150 degrees   or change 9 - 6 to 3   ??? 


 It is not the addition and subtraction property.    Recall those require add or subtract of the same or equal items on both sides of the equal mark. 

 

  WHEN we replace an expression with what it is equal to then we call that substitution.    8+7 can be replaced with 15 because we know they are equal.

When we change the  x-2+2     to x ,   

or  when we change the   -6+2 to - 4   

or when I change the 80 degrees+70 degrees to 150 degrees  

     then we are using the 5thproperty,  Substitution Property.

 

You rarely hear a teacher say substitution in algebra, but we used it all the time!!!  We will use it considerably in geometry.   When we replace an item with its equal we are using the substitution property.   We can replace segments and angles that are congruent too.

 

 

      How do we use SUBSTITUTION in GEOMETRY???   

                  WE will see these 3 examples  many times.   

 

Be sure you understand them.  

Please write these in you notebook.  You will need to understand these for the game below and the assignment.

 

    

 

THREE more Properties to KNOW:    REFLEXIVE,  Symmetric,  Transitive.

 

 HERE is the BIG GREEN BOX I told you to know well and to write into your notes  WE need this for PROOFS.

 

In algebra, the REFLEXIVE PROPERTY is rarely spoken of, but it too will be used often in Geometry. The best way to explain it is with an example.   

 

Reflexive property example:

If two figures , say a RECTANGLE and a SQUARE, are drawn such that one segment or one angle is fully shared by the 2 figures then we use the reflexive property to state that those shared parts are equal or congruent. Consider the diagram:

 LOOK at long rectangle ADEB and square AFCB.  

Segment CB is shared by both of them; angle A is shared also.   

 

Segment CB and angle A are in both the rectangle ADEB and square AFCB.

    Be sure you look at the diagram closely; that is the only way to know if something is fully shared.

 

CB is the same segment (and same length) whether you are considering rectangle ADEB or square AFCB.

Angle A is the same angle whether you are referring to rectangle ADEB or square AFCB.   Look at the diagram to know this.

 

REFLEXIVE  PROPERTY  see picture here:

 

So when a line segment or angle are shared by two or more figures then we may need to use the REFLEXIVE  PROPERTY in our proofs. 

 

 ***********************************************************************************

A  little review:  

It would be incorrect to say angle C is congruent to angle C for the diagram above, because of what we studied in Chapter 1.

Angle C is ambiguous., unclear.

Angle C does not clearly tell me what angle to consider.  There are 3 angles at C:  straight angle BCE and right angle BCF and the right angle FCE.   Hope you take the time to look at those.   Do you see them??

 

 

 

The SYMMETRIC PROPERTY is occasionally used in algebra and occasionally in geometry.


For algebra we know that 2x = 6 is really the same as 6 = 2x.

   We can trade left side and right side of the equation.

 

 For geometry we can do the same with congruent items or equal items.

 

 If segment AB is congruent to segment CD   THEN 

we can also write segment CD is congruent to segment AB.

 

 

 Or angle B is congruent to angle M THEN angle M is congruent to angle B.

 

These illustrate the symmetric property. 

 

 ********************************************************************************************

The TRANSITIVE PROPERTY requires 2 equations  

                                      and they each have a similar side .

If A=B and B=C,  then our conclusion is A=C.   Two equations and both have a side of "B".

 

If 2x +3x  =  r - t     and  r - t  = 30 

   Two equations and both have a side of "r-t". 

 THEN our conclusion is 2x +3x  = 30 by transitive property

I am illustrating the transitive property , Both equations have the  r- t.  .


The first equation :          2x +3x r  - t.

The second equation :    r - t.  = 30 .       Look closely.   

WE must have the two equations  to use the transitive property.

                 THEN our conclusion is 2x +3x  = 30 by transitive property.


We will use all of these properties as REASONS in our upcoming work.

 ***************************************************************************

 Now for geometry,:

If / A is congruent to / B                              

   and / B is congruent to / M  

then we know / A is congruent to / M by the transitive property

( angle A= angle B and Angle B= angle M then angle A= angle M. )

 

 

The properties should all be added to your Geometry Collection.

    You will use them soon to JUSTIFY your steps of work in your Assignment problems.

 

PLEASE WATCH this video.   IF you have trouble let me know.  Be sure to tell which class and lesson.

Here is a link to it at YouTube:  http://www.youtube.com/watch?v=zQ6Y0fismDg

 

 

 

 

 

LET us take a break  and play a game.

 

  It will help you look for the details of the properties.     

LOOK at the "THEN" part of the statements.

 

 You must know the REASON for the "THEN" part of the statements. 

 

You may start the game over at any time.  

It might take a few minutes to figure it out.     


Let me know how you do.   

   Properties of Equality for algebra and geometry -LEVEL 1rr:  

CLICK this link....  http://www.quia.com/rr/366508.html

 

 

Now study Example 1 on page 38.    

Please look at that example before continuing to read here.

 STOP and come back.

 

 

You will use these PROPERTIES soon to JUSTIFY your steps of work in your Assignment problems.

 

Be sure you read the REASONS  for each step.  

That is what you are learning to do here is write the REASONS.   That means you are JUSTIFYING your steps of work. 

 

REASONS  in a geometry class must be a

      1.  Postulate or

      2.  Property or

      3.  Definition from you Geometry Collection.

      4.    Later we will also use Theorems and Corollaries... 

 I hope you have your Geometry Collection.   You must have it.

 

************************************************************************

Can you tell how they multiplied the

1)      3x = 6  - 1x/2 .

 

Hope you see there are 3 terms and that each was multiplied by 2 to get:

2)      6x = 12 - 1x .   That was using MULTIPLICATION PROPERTY above.

 

Now for step 3), the reason of addition was given. What did they add?  

 They added 1x to both sides of the equation.

3)       7x= 12.   That  was the ADDITION PROPERTY  

                      -----  ADD same or equal items to BOTH SIDES.

 And for step 4:

4)  They divided both sides of 7x=12 by the 7.

                                SEE   7x/7 =12/7   that was using the Division Property.                                  We have  x= 12/7.

 

 

 

  OK, it is time for another game similar to the other,

but at a more advanced level.

 

It will use 6 of the properties rather than just 3....


Please play until you can win the grand prize.  I call it my level 3 game.
Properties of Equality for Algebra and Geometry(2-2) -LEVEL 3rr   

         http://www.quia.com/rr/370138.html 

 

Assignment GAME 2-2.  Take a screenshot of your win.

You can use SNIPPING TOOL on Windows computers   or  I use a program called Screenhunter (I love it).  It is a free download.

If you use PrtScr  button on your computer then follow the directions on HERE:   Inserting Screen Shots

 

Also, let me know in your assignment how many times you played. Write it down for your assignment above.

 

THERE is a second part to this LESSON.  It is in the Navigator box.

 

    You now