Explain how two triangles can have 5 parts (sides and angles) of one triangle congruent to 5 parts of the other triangle and yet, the two triangles are NOT congruent!



Solution:

The triangles are similar to each other.

The key to this problem is that the 5 parts that are congruent in the 2 triangles are not the corresponding parts. Otherwise, the triangles would be congruent by SSS or SAS...

All three pairs of angles must be congruent.
The three pairs of sides can not be congruent or else it would be SSS.

Here is an example of how this can work:
Triangle ABC is similar to triangle DEF.


       Triangle ABC     Triangle DEF
       ------------     ------------
       angle A       =  angle D
       angle B       =  angle E
       angle C       =  angle F
       side AB = 18     side EF = 18   
       side AC = 12     side DE = 12   
       side BC = 27     side DF =  8
Thus, five pairs of parts are congruent, but the sixth pair is not congruent.
The ratio of the the three pairs of corresponding sides is 3:2. 

        AB   BC   AC      18   27   12        
        -- = -- = --  or  -- = -- = --  
        DE   EF   DF      12   18    8 

                             3
        which all reduce to ---
                             2


Correctly solved by:

1. Jon Pence Winchester, VA