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### 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.

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
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

Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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