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Lesson2-6

Page history last edited by Math in a Box - Susan Johnsey gm 1 year, 3 months ago

Lesson 2-6  Planning a Proof       part 1

There is quiz to take at this time.  

You will need your Geometry Collection and understand how to find the complement or supplement of an angle.    

 

The quiz should take only 10 to 15 minutes and is worth 30 points.  Then return here to complete this lesson. 

You may take NO more than 25 minutes to complete the test.  You MUST have your user id and password for the site QUIA.  It is in your folder, look for QUIA INFO.

 

Complete ."A QUIZ for 2-5 and 2-6",  see it in your student folder in the Navigator box.   Then return here to complete this lesson.

 

YOU MUST READ AND write notes from THIS LESSON.  

  I HAVE MANY COMMENTS TO MAKE.  

YOU WILL STRUGGLE FOR MANY CHAPTERS IF YOU DO NOT UNDERSTAND THIS LESSON. 

 

GEOMETRY is about LOGIC and following "rules" to prove an idea is true. 

      YOU MUST:

  1.           Learn to be logical.
  2.           Look carefully at DETAILS in the words.
  3.           Look carefully at DETAILS in the diagrams.
  4.           Present your proof ONE STEP at a time  using your GEOMETRY COLLECTION. 
    1. You will need it greatly so I hope you have yours up to date.

 

  YOU MAY TAKE TWO DAYS FOR THIS LESSON.   

 

  YOUR CHAPTER TEST IS EXPLAINED  AT THE END OF THE LESSON 2-6 part 2.

 

 

Add the 2 theorems to your Geometry Collection:

              Theorem 2-7 and Theorem 2-8 .

 

I will give you special names for these at the end of the lesson.  

Be sure to add these names to the theorems. 

 

 

 

If angle B = 30 degrees and angle C = 60 degrees then we know that they are complementary. 

Right?  if we add angle B to angle C we have 30 + 60 = 90 degrees.  

 

 Thus these two angles are complementary for that is the definition of complementary angles.

 

Then if I tell you angle D is also complementary to angle B.

We now have angle D and angle C that are each complementary to angle B.  What can you tell me about angle D? 

There are 2 very obvious things.

         1)       Angle D = 60 degrees since angle B = 30 degrees.

        2)      Angle D  is congruent to angle C.       THEY both measure 60 degrees.

 

The above is an example for Theorem 2-8.

   See how I started with 2 angles then moved to 3 angles: B, C and D.

If you did not get that then READ it AGAIN because I am about to go to 4 angles!

 

 

We are adding two angles to create 90 degrees for complementary.   

 I used 3 angles:    B, C, and D above and gave you their degree measure. 

 

Look at these 4 angles.

Now we will add two angles to create 180 degrees for the supplementary.

 

Theorem 2-7 makes a similar statement about supplementary angles (2 angles sum = 180).

 

 

Here are the SPECIAL names for these Theorems.   Please learn these.

 

  I will refer to Theorem 2-7 as the Supplementary Angles Theorem.

  I will refer to Theorem 2-8 as the Complementary Angles Theorem.

 WRITE these names in your geometry collection.

 

Now add these names to theorem 2-7 and 2-8 in your Geometry Collection.  

      YOU must use the NAMES above in your proofs and work.

 

 

     If you need a good diagram for both of these then click this link.  

 

The other ideas presented in this lesson concern "Writing PROOFS".

 

 HERE are the 6 parts.  

           I am using the Complementary Angles Theorem as an example.

 

 

 Part 1 "state the theorem"

         For Complementary Theorem:  If two angles are complements of (2)congruent angles

                                           then the two angles are congruent.

 

Part 2  This is the "GIVEN".   It is the first part of Part 1 or the " if  "part of the theorem.   Do you see it above? 

For Complementary  Angles Theorem:   two angles are complements of 2 congruent angles.  

Let's give them names

     angle A and angle B are complements     

                                             TO    the congruent angles C and D. 

Or we could say:

    GIVEN:  angle A is complementary to angle C and angle B is complementary to angle D  

                     and  angle C and angle D are congruent.

 

 

Part 3  This is the "PROVE".   It is the  "then" part of the theorem.   Do you see it above? 

 

      For Complementary Theorem:  "Then the two angles( the first ones mentioned in the theorem) angle A and angle B  are congruent." 

 

    PROVE:  angle A is congruent to angle B.

 

Part 4   "Draw a diagram"

 

To do this you must read the GIVEN and the PROVE and imagine what it is saying and then draw and label it.   

Look back at Part 1.  We have two angles that are complementary to 2 other angles. 

For Complementary Angle Theorem: 

There are 4 angles that we need to draw.

See my diagram above  that we have already studied.

 

 

Part 5 is the list of geometry statements.

 

 

Part 6 is the list of reasons that belong with each statement in Part 5.

Reasons are

  • o   definitions or
  • o   properties, or
  • o   postulates or
  • o   theorems
 REASONS are NOT your comments or thoughts.

 

Look at page 61, Theorem 2-7.   Do you see all 6 parts?

 

Now look at page 64, problem 17.   One of the parts is missing!!!   Which one??

I do not mean part 6 , the reasons.   The reasons are numbered and have blanks for you to fill.   There is a part that is not there at all.   Do you know yet?   Email me your answer.

 

You will learn to read the GIVEN  ("if") and PROVE  ("then") for several theorems 

and then write the statement and reasons.    

 

The exercises in this lesson will help you learn to write the REASONS,.

The diagram will be drawn for you in most cases, but sometimes you will have to draw it. 

 

So here is my big hint.   You should always be able to do the first Statement and its Reason, for they are GIVEN to you  in the problem,  sometime written, but sometimes as a drawing.  Practice finding the GiVEN ( look at if-part of statement and the diagram markings).

 

See the first statement and reason for the proof on page 61 and page 44.  

 It is the GIVEN (IF) part of the statement we are proving!!!     

You need all the info that is GIVEN  and the diagram.  

These are the vital building materials you must have to begin construction of a proof.

 

 Now look at the theorems again on page 61 and 44.    Open your book.

          What do you notice about the  LAST Statement ?   It is the PROVE (then-part) statement, or part 3.  

           The Reason for this last statement will have to be determined; you must figure it out!

But, it will be a definition or postulate or property or theorem. 

 

           CLUE:  LOOK at the next-to-last statement and LOOK at your GEOMETRY COLLECTION.

 

 Know the 6 PARTS,

   and know the FIRST Statement is usually your GIVEN statement with "GIVEN" as the reason,

   and know the LAST statement is from the Prove statement - you must decide its reason. 

Knowing the 3 lines above will greatly increase your ability to write a correct proof.

 

Before you begin these please review your Geometry Collection.  

Look for the

  • Angle Addition postulate, 
  • Reflexive property,
  • Substitution property, see example in brown box below.
  • Subtraction property,
  • Division property.  

 

You will need your Geometry Collection to complete these assignments.

 

 

Look at page 61,  Reason 3 of the proof states the Substitution property. 

Angles 1 and 2 are supplementary.    Angles 3 and 4 are supplementary.

 

Can you tell me what was substituted, see below?   We will use the Substitution property often.

 

 LOOK at step 2 to understand step 3.               page 61

 

     2 m angle 1 + m angle 2=180           m angle 3 + m angle 4 = 180    

 

                 Notice that both the "yellow" and the "green" = 180.  

                Would not that mean the "yellow" equals  "green"?           

                                          sure it does.   So that is our step 3.    This is substitution . 

                You can think REPLACE the first 180 above with the "green".  That gives us:

 

      3. THUS m angle 1 + m angle 2  =  m angle 3 + m angle 4           Substitution property.

  

 

Do Written Exercises 1 to 21 odd page 63  to 64.

 I hope you are still completing and checking all of your work.  It is worth it, you know.

 

Let me know which of these you missed and what your question is.

 

REMEMBER at THE bottom of LESSON 2-1  IS A VIDEO WHERE I SHOWED YOU HOW TO USE THE PROPERTIES AND POSTULATES TO WRITE A PROOF.   YOU MAY WANT TO WATCH IT AGAIN.    IN THIS LESSON WE ALSO USED THEOREMS AND DEFINITIONS.

 

RECALL how to find the LISTING of Assignments?   You learned that in Lesson 2-5 part 2 when you found Assignment 2-5B.   Search for the listing part 1.

 Do Assignment 2-6A 

 

I have divided this lesson into two parts.   

After you finish the assignment above read the Lesson 2-6 second part.

 

Let me know your questions any time.  I answer quickly most days.  Email again any time you are waiting.