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```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.

Given: rhombus ABCD with diagonal AC
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. _________________________

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
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/
```
Associated Topics:
High School Triangles and Other Polygons

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