Associated Topics || Dr. Math Home || Search Dr. Math

### 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

Have fun drawing a conclusion!

-Doctor Mateo,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
High School Trigonometry
Middle School Geometry
Middle School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search