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lf the sum of one of the base angIes and the vertex angIe is the same for two different isosceIes triangIes, must the triangIes be simiIar?

Apr 21, 2021

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Mmmm....this is an  interesting question

Let  the  base angles of  both triangles  be  equal

And let's assume that we  can create  similar triangles

Let  the  base  angle  in one triangle  = x and  the apex angle = y

Their sum is   x  +  y

Now   let  the  base  angle  of  another  triangle = (x - a)   and let  the apex angle = (y + a)   where  a is positive

Their sum  is also  x +  y

In the first   triangle    2x  +  y   =180

In the  second  triangle

2 ( x - a)  + ( y + a )    =180     simplify

2x  + y   - a   = 180

2x  + y  =  180 +  a

Since  a  is  positive....2x + y   in one  case  =180    and in the other case it is more  than 180

This is impossible.....so.....since  the  final result leads to an  absurdity.....our  assumption must  be false....

So....the  triangles cannot  be made similar    by   having  the  same  sums  for  one base angle and  the apex angle

Apr 21, 2021
edited by CPhill  Apr 21, 2021
edited by CPhill  Apr 21, 2021