Parallelogram and Isosceles Triangle

 

Definition ( Parallelogram )

                  For a quadrilateral ABCD, if   //  and  // ,

                  then  ABCD is known as a parallelogram.

 

                              

 

Property 1:     B =  D and   A =  C .

              Proof:

                              1.    //  

                                    So,    A+  B=180⁰

                              2.   //

                                    So,   A+  D = 180⁰

                              3. From 1,2, we have

                                            B =  D

                              4. Using similar approach,   A=  C

 

Property 2:    =  and  =

 

              Proof:

 

                               

1.      Construct   as shown in the diagram above.

2.       //  and  //

So,   1=  2 and  3=  4 .

                         3.   1=  2,  = ,  3=  4

                                 ABC  CDA   ( ASA )

                                  =  and  =

 

 

 

 

 

 

 

 

Theorem :  For a quadrilateral ABCD, if   //  and  = ,

                   then ABCD is a parallelogram.

                               

 

         Proof:

                                  

 

1.      Construct  as above. Then

          1 =  2  ( because  //  )

2.   = ,  1=  2,  =

       ABC  CDA  ( SAS )

       BCA =  DAC

    So,   //  

3.  //  and  //

     So,  ABCD is a parallelogram.

 

 

Theorem:    If  quadrilateral ABCD is a parallelogram, then 

                         =  and   =  .

 

                           

                 Proof:

1.      From the previous theorem, ABCD is a parallelogram

        =  ,  and  1 =  2 .

     Furthermore, we have    AEB =  CED

    So,    AEB   CED

2.    AEB   CED

          = ,   =

 

 

 

 

 

Definition ( Isosceles Triangle )

                   For a  ABC, if  = , then we say that this triangle is Isosceles Triangle.

 

                  

 

 

 

 

 

Property 1:     B =  C if and only if   = .

 

                Proof:

                          (  ): given  = , let’s prove  B =  C  .

                                    1.    = ,  A=  A,  =

                                            ABC   ACB  ( SAS )

                                          Thus,   B =  C .

 

 

                          (  ): given  B =  C, prove  =

                                    1.   B =  C,   = ,  C =  B

                                            BAC    CAB  (ASA)

                                          So,   =  .

 

    From above, we know that   B =  C if and only if   = .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Theorem ( Median of Isosceles Triangle )

                      If   = ,   = , then     .

                              

                   Proof:

 

                                 1.   = ,  = ,  =

                                        ABM   ACM ( SSS )

                                        BMA =  CMA

 

          2.  Furthermore,   BMA +  CMA = 180⁰ .

                 BMA = 90⁰ ,  CMA = 90⁰

              In other words,       .