Larger Angle corresponding to Larger Side in Triangle

 

Motivation:

                         In  ABC,  if   C >  B, can we say   > ?

 

                                 

                         The answer is Yes.  And we are going prove it by using the congruence theorems of triangle.

 

 

 

 

 

 

Theorem ( Larger Angle corresponding to Larger Side in Triangle )

                   In  ABC,  if   C >  B, then   > .

                              

 

         Proof:

                            1.   C >  B.  Thus, it is possible to construct an angle

                                  equal to  B within  C .   Let’s construct such an

                                  angle as shown in the diagram above.  Then

                                

                                      B =  DCB

 

                            2.   B =  DCB

                                   =

 

3.      In  ADC,  we have   +  >  .

       +  >

       >

 

 

 

 

 

 

 

 

 

Theorem ( Arc and Chord )

                         A, B, C, D are on the same circle as shown below.

                         If  arc AB is equal to arc CD,  then   = .

 

                                

 

           Proof:

1.      Construct , , ,  .

The same arc length is with the same central angle (  recall the definition of “angle” )

  AOB =  COD

 

                         2.   = ,  AOB =  COD,  =

                                  AOB   COD ( SAS )

                                 =

 

 

 

 

 

 

 

 

 

 

Theorem : Within the same circle,  larger arc is with larger chord.

                             

                                                     

 

                    Proof:

                                   If arc CD is larger than arc AB, we are going to prove

                                              > .

1.      From point D, construct a chord  such that

             =

2.      Construct .

3.      Larger arc is with larger central angle.

       the corresponding inscribed angle is also larger.

     Thus, within  CED,  we have

         E >  C

 

       >

       >