Learning to Read ProofsDate: 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/ |
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