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Math Help - Multiple Answers for a Geometric Proof

  1. #1
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    Multiple Answers for a Geometric Proof

    I am currently teaching myself Geometry using only the McDougal Littel textbook simply titled Geometry. I am doing this before going over Algebra 1 again because I feel that I still have a decent enough grasp of Algebra to deal with whatever may be needed for Geometry. I just got the the section on Geometric Proofs which started with a basic review of proofs for equations, which I found rather simple after reviewing my Properties of Equality. However once I got to applying proofs to Geometric problems, I have hit the first really hard topic for me, seeing as I am my only teacher. I managed to get an understanding of the material after a few hours of work. Then, I tried to write out the proof for the following problem:

    (See link for picture first)
    2011-06-05_203844.jpg picture by MDS1005 - Photobucket



    Given: RQ = TP
    ZQ = ZP
    Prove: RZ = TZ

    Now, I worked through this, thought I had gotten a correct answer, and almost screamed when I checked the answers in the back of the book and I was wrong.

    Here's what I came up with

    Statement Reason
    1 RQ= TP
    ZQ = ZP Given

    2. RQ-ZQ = TP-ZP Subtraction Property of Equality

    3. RZ + ZQ = RQ
    TZ + ZP = TP Segment Addition Postulate

    Now, at this point, in my head, I used the Substitution Property of Equality to formulate the following equation:

    RZ + ZQ - ZQ = TZ + ZP - ZP

    Simplifying this gave me my final statement of

    4. RZ = TZ Substitution Property of Equality


    Now, this was no where near the answer given in the textbook, although it makes sense to me. It has one or two fewer steps than the one given. The only thing I can see that might be wrong, taking into account my limited experience with proofs, is the fact that I am trying to relate a step to one that is not immediately before it in the sequence. Can anyone tell me if this is completely wrong, or if it is an acceptable proof?
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  2. #2
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    I would just say that since \displaystyle RQ = TP and \displaystyle ZQ = ZP, then

    \displaystyle \begin{align*} RZ &= RQ - ZQ \\ &= TP - ZQ\\ &= TP - ZP\\ &= TZ \end{align*}

    Q.E.D.
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  3. #3
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    In mathematics I am likely to see in the future, is the two column proof a pretty standard model for proofs? I've heard differing answers to this. Secondly, I've heard a few people mention that proofs are the hardest thing a student will encounter in their first geometry class. Any opinions on this? Also, I'm pretty sure now that the proof I provided is valid. Confirmation of this would make me feel a lot better about moving on in my studies however.
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  4. #4
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    Quote Originally Posted by MDS1005 View Post
    In mathematics I am likely to see in the future, is the two column proof a pretty standard model for proofs? I've heard differing answers to this. Secondly, I've heard a few people mention that proofs are the hardest thing a student will encounter in their first geometry class. Any opinions on this? Also, I'm pretty sure now that the proof I provided is valid. Confirmation of this would make me feel a lot better about moving on in my studies however.
    You are not likely to see two-column proofs in the future.

    Proofs are "hard" because teachers historically suck at teaching them. Their approach is usually, "You get better at them by doing a bunch of them."
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