Proof of Congruency

Date: 10/13/96 at 19:23:15
From: R. J. Ledoux
Subject: Congruent Lines

I am lost in geometry, especially with proofs.

 	Given: line PR bisects <QPS and <QRS
    Prove: Segment RQ is congruent to segment RS

                        P
                        .
                       /|\
                      / | \
                     /  |  \
                    /   |   \
                 Q .    |    . S
                    \   |   /
                     \  |  /
                      \ | /
                       \|/
                        .
                        R
                                                           

Date: 10/17/96 at 10:13:9
From: Doctor Wilkinson
Subject: Re: Geometry

Problems like this one generally fall into a pattern.  You'll be given
a picture or diagram and some information about it, usually that some
parts of it are equal or congruent.  You're asked to prove that some
other parts of it are equal or congruent.  

In your case, you're given that PR bisects the angles QRS and QPS.  
"Bisect" means "divide into two equal parts."  So what you're given is 
the information that angles QRP and PRS are equal, and also that 
angles QPR and RPS are equal.

You're asked to prove that the line segments RQ and RS are congruent.
What do you have to work with?  You have theorems and axioms which 
apply in certain situations.  In the problems you're looking at these 
are theorems and axioms relating to congruent triangles.  

The pattern:

    (1) This triangle and that triangle have certain parts congruent.
           (Given information)

    (2) Therefore the triangles are congruent.
           (Theorem or axiom)

    (3) Therefore the other corresponding parts are congruent.

In your example there are just two triangles apparent, so you should 
try to show they are congruent.  You know that the angles QRP and PRS 
are equal and that the angles QPR and RPS are equal.

Now, you have three theorems (they were theorems when I took high-
school geometry, but some people call them axioms nowadays; in any 
case you can use them any time you need them).  They're often 
abbreviated SAS (side-angle-side), ASA (angle-side-angle), and SSS 
(side-side-side), for the parts that are known to be congruent.  

In your case you're off to a good start for applying angle-side-angle, 
since you know that two angles of one triangle are congruent to two 
corresponding angles of the other triangle. But notice that you 
always need at least one side to apply any of these theorems. What 
sides are available? Well, one triangle has sides QP, PR, and RQ.  
The other triangle has sides PR, PS, and SR.  PR is a side of both 
triangles. You can't get more congruent than that. So we can apply 
the angle-side-angle theorem and conclude that the two triangles are 
congruent.  

Now step (3) in the pattern above shows that the segments RW and RS 
are congruent.

I hope this helps.  Feel free to ask follow-up questions.

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