Congruence and Triangles

Date: 12/13/97 at 20:15:56
From: Frank Soarees
Subject: Congruence and Triangles

Can you please explain how to determine, using SSS, SAS, and ASA, how 
a shape is congruent or not and how and when to put the triangle 
congruence properties?  I have tried but I just don't understand it.  
Please start from the beginning. Thanks.

Date: 12/13/97 at 21:22:55
From: Doctor Guy
Subject: Re: Congruence and Triangles

First of all, let us agree on a few things:

(1) two line segments are congruent if, and only if, they have 
    the same length. This means that if we know that segment FI 
    is 7.3 cm long and so is segment SE, then segment FI is congruent 
    to segment SE. We can write FI = SE here, because I cannot type a 
    congruence sign, which is like an equals sign (=) with a tilde 
    (~) over it.

(2) two angles are congruent if, and only if, they measure the same 
    number of degrees. So if angle GUS measures 81 degrees and so does 
    angle JON, then angle GUS and angle JON are congruent. The best I 
    can do here is write <GUS = <JON, because of the limitations of 
    e-mail.

(3) congruence of segments and angles is reflexive, transitive, and 
    symmetric. Meaning, segment AB is congruent to itself; and if 
    segment CD is congruent to segment EF, and segment EF is congruent 
    to segment GH, then segment CD is congruent to segment GH; and if 
    segment IJ is congruent to segment KL, then segment KL is 
    congruent to segment IJ. The same holds true for angles.

(4) two triangles  are congruent if, and only if, they have 3 pairs 
    of congruent sides and three pairs of congruent angles. In other 
    words, triangle ABC and triangle XYZ are congruent if, and only 
    if, angle A is congruent to angle X, angle B is congruent to 
    angle Y, angle C is congruent to angle Z, side XY is congruent to 
    side AB, side BC is congruent to side YZ, and side AC is congruent 
    to side XZ. 

    Note that when you say two triangles are congruent, you have to 
    get the letters in the proper order to show what's congruent to 
    what. If I had said triangle BAC was congruent to triangle YXZ, 
    then that's okay (check it out). But if I say that triangle YZX 
    is congruent to triangle CAB, then that's wrong. (Why?)

               A *                          X *




         B *--------------* C       Y *------------------*Z

(I'm not going to draw the diagonal lines. You can fill them in 
yourself.)

Note that there is a LOT of things that you have to check in that last 
one. Six pairs of things have to be congruent. Wow. Most of the time 
you don't have the luxury of having all that information. Are there 
any shortcuts?

It turns out that yes, there are some shortcuts, and those are the 
SSS, SAS, and ASA postulates that you mentioned.

SSS: the letters stand for "Side-Side-Side". What that means is, if 
you have two triangles, and you can show that the three pairs of 
corresponding sides are congruent, then the two triangles are 
congruent. This is a postulate, not a theorem, meaning that it cannot 
be proved, but it appears to be true so everybody accepts it. So if 
you have triangle ANT and FLY, and you can show that AN = FL, NT = LY, 
and AT = FY, then the two triangles are congruent.

SAS: the letters stand for "Side-Angle-Side". What that means is, if 
you have two triangles, call them BOY and GRL, and you can find two 
pairs of congruent corresponding sides AND a pair of congruent 
corresponding angles that is BETWEEN the two pairs of sides we just 
mentioned, then the two triangles are congruent. Again, this is a 
postulate.

Suppose in triangles BOY and GRL we know that BO=RL, BY=GR, and <B=<R. 
I claim that triangle BOY is congruent to triangle RLG, using the SAS 
postulate.

ASA: the letters stand for "Angle-Side-Angle". What that means is, if 
you have two triangles, call them DOG and CAT, and you can find two 
pairs of congruent corresponding angles AND a pair of congruent 
corresponding sides that is BETWEEN the two pairs of angles we just 
mentioned, then the two triangles are congruent. Again, this is a 
postulate.

Suppose in triangles DOG and CAT we know that <D = <T, <O = <C, and 
DO = CT. I claim that triangle DOG is congruent to triangle TCA, using 
the ASA postulate.

You may be wondering if there is a SSA postulate. No, unfortunately, 
the SSA (or ASS) postulate doesn't always work, especially in acute 
triangles. There is no AAA congruence postulate either, since the two 
triangles would have the same general shape (i.e. be similar) but one 
might be much bigger than the other. There is an AAS theorem, but 
let's not worry about that right now.

If the sides and angles do not correspond, there is no congruence. 
For example: if we have triangle NBC and triangle KLM, and <N = <K, 
<B = <L, and KL = CN, then those triangles are NOT necessarily 
congruent.

Here are four exercises for you to try. I suggest you draw a picture.
(The answers are down below. Hide them from yourself.)

1. In triangle RUN and triangle HID, <R = <D, <U = <I, and RU = DI. 
   What triangles are congruent, if any, and why?


2. In triangle FRE and SLV, FR = LV, EF = SL, and <F = <S. What 
   triangles are congruent, if any, and why?


3. In triangle MUS and CHR, <S = <H, US = HR, and <U = <R. What 
   triangles are congruent, if any, and why?


4. In triangle QWE and RTY, QW = TY, WE = RY, and QE = RT. What 
   triangles are congruent, if any, and why?

I'm leaving some blank space here.






here's some more.








Here are the answers:



4. (Yes; triangle QWE = triangle TYR by SSS)
3. (Yes; triangle SUM = triangle HRC by ASA)
2. (No; the sides and angles do not match)
1. (Yes; triangle RUN = triangle DIH by ASA)

I hope this helps a little. Sorry it's so hard to draw pictures on 
here.

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

Date: 12/14/97 at 19:00:24
From: Franklin J. Soares
Subject: Thank You!!

Hi!  I wrote you a question the other day and you answered it 
immediately. I am impressed! Your information was very very helpful.  
You explained everything clearly and completely. I like your method of 
explaining things, the way you give a definition and explain it in a 
complete manner. Thank you very very very much. Keep on doing what you 
do, it is really worth it.

- Franklin Soares