Congruent Triangles in a Rectangle
Date: 11/11/1999 at 15:15:28 From: Lacey Subject: Congruent Triangles (Proof) Dr. Math - I've been working on this problem for my Geometry class for days now. I'm not very good at proofs, so I am struggling to even get started. Here is the information: Given: AB is parallel to RT AR is perpendicular to AB BT is perpendicular to RT AB is congruent to RT AR is congruent to TB Prove: Triangle ABR is congruent to Triangle TRB (The picture is a rectangle with a diagonal line from corner R to corner B going up. Point A is in the upper left hand corner, R is bottom left, B is upper right, and T is bottom right. T and A equal 90 degrees.) I hope you have enough information. I am currently trying to get the various postulates straight from each other; the Side by Side Postulate, Side-Angle-Side Postulate, and the Angle-Side-Angle Postulate. Any ideas on how I can remember them? Thanks a bunch! Lacey
Date: 11/11/1999 at 17:11:56 From: Doctor Peterson Subject: Re: Congruent Triangles (Proof) Hi, Lacey. You gave me wonderful information! You described the picture perfectly, which shows me you have what it takes to prove congruence, because you know how to describe something accurately enough so I can draw it for myself. That's exactly what a congruence proof is all about. It sounds like what you really need is to get familiar with SSS, SAS, and ASA so you can recognize them and know how to use them. Let's focus on that rather than on this particular problem. First, take a look at the first part of this answer from our archives: Theorems for Quadrilaterals http://mathforum.org/dr.math/problems/bethmarie.11.12.99.html It describes the three congruence theorems. You can remember which ways work by just picturing how you can construct them, or by just remembering you have to have either all the sides (SSS), or one thing and the two on either side of it (SAS or ASA). To put it another way, you always need three things, and either one, two, or three of them can be sides - call them the one-stick, two-stick, and three-stick methods of building a triangle. Just make sure the angle is in the right place if you have only one of them. Now how do you apply this? One way is to redraw (or re-imagine) the two triangles in the same position, so you can more easily recognize what goes together. Here's your original problem: A B +---------+ | / | | / | | / | | / | +---------+ R T Here are the two triangles pulled apart: A B B +---------+ + | / / | | / / | | / / | | / / | + +---------+ R R T And here I've turned ABR around so they're facing the same way: R B + + / | / | / | / | / | / | / | / | +---------+ +---------+ B A R T Notice that the letters of ABR and TRB match: A and T are in the same places, and so on. They were named in this order to show which pairs of vertices go together. Your "givens" are actually far more than you need to prove the conclusion. You could use any of SSS, SAS, or ASA to do it. Let's list the parts of the two triangles and what you know about them: ABR TRB --- --- Sides AB = TR (and parallel, too) BR = RB (in fact, they're the same segment) RA = BT Angles RAB = BTR (both right angles) ABR = TRB (alternate interior angles) BRA RBT (no immediate relationship) There are not 3, but 5 things you can see immediately or very easily to be the same. See if you can find three that prove congruence. (Hint: SSS is probably the easiest.) - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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