Learning to Read Proofs

Date: 10/31/2003 at 14:23:27
From: Beth
Subject: SSS ASA SAS Proofs

I sort of understand the principles of SSS ASA SAS Proofs but I never
know what comes next in the proof.

One question that I had to answer was:

  Given: rhombus ABCD with diagonal AC
  Prove: AC Bisects <BAD
  Proof: 
  1. A rhombus is an equilateral quadrilateral
  2. CB is congruent (=) to CD
  3. _________________________
  4. AC is common to triangle CBA and triangle CAD
  5. Triangle CBA = CDA by the SSS theorem for congruent triangles
  6. _________________________
  7. Therefore AC Bisects <BAD

The blanks are steps that I was supposed to fill in.  My answers were
marked wrong, and given the correct answers I can see how they make
sense, but I can never seem to get the answers on my own.

Date: 10/31/2003 at 15:47:50
From: Doctor Peterson
Subject: Re: SSS ASA SAS Proofs

Hi, Beth.

If you haven't seen our FAQ on this topic, I think it will help:

  Proofs
    http://mathforum.org/dr.math/faq/faq.proof.html 

I think the key to this kind of problem, where you are filling in 
steps in someone else's proof, is NOT to think of it as "what comes 
next", but rather "what fills the gap". Don't try to figure out step 
3 based only on what comes before, because there may be several ways 
to do the proof, and certainly there are different orders in which 
you can put some of the steps. What you need to do is to read through 
what they give you and ask, "what is this step dependent on?" at each 
step.

In your example, step 5 uses the SSS theorem, so it needs to know 
that three pairs of sides are congruent. In steps 2 and 4 two of 
those pairs are shown to be congruent; the "gap" you have to fill is 
to show that the OTHER pair are congruent. So you write in AB=AD, and 
figure out why that is true. 

Similarly, step 6 has to fill the gap between a statement 
about congruent triangles, and one about a bisected angle. What does 
it take to say that an angle is bisected? You have to know that the 
two parts are congruent. How can you show that two angles are 
congruent? By showing that they are corresponding angles in a pair of 
congruent triangles. So the gap is filled by a statement that 
<DAC = <BAC.

Do you get the idea? This is not a test of the skill you need for 
writing your own proofs, but of an important skill in READING proofs: 
following the reasoning and filling in gaps. You almost have to read 
backward in order to do this; in reading a proof, you are always 
looking back and asking, "Where was the information he needed in 
order to be able to say this?".

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/