Proving Lines Congruent
Date: 03/29/2002 at 14:01:50 From: Allison Subject: Geometry Proofs Dr. Math, I am having a little trouble understanting proofs in geometry. One in particular that I am having trouble with is as follows: I hope you can understand the drawing. The picture is of a slanted rectangle with A and B on the bottom; A and D is the left side, and B and C is the right side, and D and C is the top. Line DB cuts the rectangle diagonally in half. L and M are on line DB. BCM forms an isosceles triangle and ADL forms another isosceles triangle. You are given that angle DAL is congruent to angle BCM angle CDL is congruent to angle ABM line DC is congruent to line BA You have to prove line AL is congruent to line CM I would greatly appreciate your help in solving this problem. Thank you, Allison
Date: 03/29/2002 at 16:49:57 From: Doctor Peterson Subject: Re: Geometry Proofs Hi, Allison. It does help a lot to have someone to talk to about proofs; they are often hard for students to grasp at first. Let's work through this one together. I presume that by "slanted rectangle" you mean that it looks like a parallelogram; and that nothing tells you explicitly that this is true, or that what look like isosceles triangles really are - these are just descriptions of what the picture looks like, for my guidance. Let me know if that is wrong. Here is my version of the picture: D z C +--------------------------------+ / \ y / / / \ / x/ / \ L / / / + / / / / \ / ? / / ? / \ / / / / + / / / M \ / /x / \ / / / y \ / +--------------------------------+ A z B I've marked "x" and "y" where pairs of angles are congruent, and z for the congruent sides. You want to prove that the two segments marked with "?" are congruent. I would start by looking at the implications of the facts you know, so we can get a feel for what I have to work with: The x's suggest that triangles DAL and BCM may be congruent, though we don't know enough yet to say they are. The y's on opposite sides of transversal BD tell us that AB and DC are parallel - that will likely be one of the steps in the proof. That and the z's tell me that this is in fact a parallelogram; you probably have a theorem or two that will prove that. Other facts arise from this, such as that AD and BC are congruent and parallel, and that opposite angles are congruent. These are all worth paying attention to! Now we can look from the other end: what would help in proving our goal? Well, if, as I suggested, we can prove that triangles DAL and BCM are congruent, that will get us to the goal. What would it take to prove that? There are several triangle congruence theorems, each of which requires knowing three parts of the triangles. What parts do we know? angle DAL = BCM - given side DA = BC - because we have a parallelogram There's another pair of angles we can prove congruent without much effort, using the facts we now have to work with. See if you can put these ideas together into a proof. If you haven't seen our FAQ yet, it has some good general ideas on how to approach proofs: http://mathforum.org/dr.math/faq/faq.proof.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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