Determining Triangle Similarity

Date: 05/26/98 at 22:31:37
From: Big Red
Subject: Trigonometry

If you have triangle abc with side ab = 65 and side cb = 52, and you 
have another triangle def with side fe = 40 and side de = 24, how can 
you tell if they are similar?

Date: 05/30/98 at 23:39:35
From: Doctor Mateo
Subject: Re: Trigonometry

Hi Big Red,

The question you ask is about similar triangles. One of the most 
important things that you need to know about similarity is what it 
means for two figures to be similar. Similarity means that:

    (i) the two figures have congruent angles and
   (ii) the sides are in proportion.

To assist you making a determination as to whether two triangles are 
similar, you might remember from geometry class some theorems about
similar triangles.  

Here are a few important ones:  

     (i) AA~ (angle-angle similarity). This one says that if you know 
         two of the angles of two triangles are congruent, then the 
         third angles of the two triangles are congruent, and the 
         triangles are similar.

    (ii) SAS~ (side-included angle-side similarity). This one says
         that if you have two sides that are in proportion (that is, 
         they reduce to the same fraction) and an included angle, then 
         the triangles are similar.

   (iii) SSS~ (side-side-side similarity). This one says that if the 
         three sides are in proportion (they reduce to the same 
         fraction), then the triangles are similar.

At this point you need to ask yourself what you know from the problem 
or what is presented in a diagram.

Let me present several examples to demonstrate the usefulness of the 
above theorems. (Here m< means "the degree of angle"):

Example 1:  triangle ABC with m<A = 80 and m<C = 30 and
            triangle VWX with m<X = 30 and m<V = 80.
 
   Similarity exists between the two triangles since m<A = m<V = 80 
   and m<C = m<X = 30. The third angle of each triangle must measure  
   70 degrees since the sum of the measures of the interior angles of 
   a triangle sum to 180 degrees. The similarity in this example is 
   then triangle ACB ~ triangle VXW by AA~ theorem. Notice how the 
   vertices match. An 80-degree angle of one triangle matches with an 
   80-degree angle of another triangle, and a 30-degree angle of one 
   triangle matches with a 30-degree angle of a second triangle.

Example 2.  triangle MNL with m<N = 90, MN = 25, and NL = 15  and
            triangle PNL with m<N = 90, PN = 9, and NL = 15.

   Since the m<N is 90 degrees in both triangles it is probably a good 
   idea to investigate the ratios formed by the segments that create 
   triangles MNL and PNL. A good strategy to employ here might be to 
   write the ratio of the shorter given side of triangle MNL (NL = 15) 
   to the shorter given side of triangle PNL (PN = 9). Then compare it 
   to the ratio of the longer given side of triangle MNL (MN = 25) to 
   the longer given side of triangle PNL (NL = 15).

   Since the included angle is known to be congruent, then the 
   triangles will be similar if these ratios are equal (which means 
   that the sides are in proportion). 

   So let's see:

      NL/PN = 15/9 = 5/3 and MN/NL = 25/15 = 5/3

   Since both ratios reduce to 5/3 and share an included congruent 
   angle, we have triangle MNL ~ triangle NLP by SAS~.


Example 3.  triangle ABC with AB = 15, BC = 10, and AC = 20 and 
            triangle FGH with FG = 16, GH = 32, and FH = 24.

   Since the only information you have is about the length of the   
   sides, we will need to consider the ratio of the lengths of the   
   sides of triangle ABC to the length of the sides of triangle FGH.

   The best strategy might be to put the lengths of the sides of the
   triangles in increasing order so that you have:
 
      triangle ABC: shortest length ? middle length ? longest length  
                    --------------- = ------------- = --------------
      triangle FGH: shortest length   middle length   longest length 

   So in example 3, we have the following:

      triangle ABC:   10   ?   15   ?   20
                     ----  =  ----  =  ----
      triangle FGH:   16       24       32 

   All of the ratios above reduce to 5/8 so the triangles are similar.
   If you do not get the same reduced fraction for all three ratios of
   sides then the triangles are not similar.

   Here triangle ABC ~ triangle HFG by SSS~.

Since you have information about two sides you may want to consider
what the third side of each of your triangles is. Then use one of
these theorems to help you draw a conclusion about your problem.  

Have fun drawing a conclusion!

-Doctor Mateo,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/