Congruence and Triangles
Date: 12/13/97 at 20:15:56 From: Frank Soarees Subject: Congruence and Triangles Can you please explain how to determine, using SSS, SAS, and ASA, how a shape is congruent or not and how and when to put the triangle congruence properties? I have tried but I just don't understand it. Please start from the beginning. Thanks.
Date: 12/13/97 at 21:22:55 From: Doctor Guy Subject: Re: Congruence and Triangles First of all, let us agree on a few things: (1) two line segments are congruent if, and only if, they have the same length. This means that if we know that segment FI is 7.3 cm long and so is segment SE, then segment FI is congruent to segment SE. We can write FI = SE here, because I cannot type a congruence sign, which is like an equals sign (=) with a tilde (~) over it. (2) two angles are congruent if, and only if, they measure the same number of degrees. So if angle GUS measures 81 degrees and so does angle JON, then angle GUS and angle JON are congruent. The best I can do here is write <GUS = <JON, because of the limitations of e-mail. (3) congruence of segments and angles is reflexive, transitive, and symmetric. Meaning, segment AB is congruent to itself; and if segment CD is congruent to segment EF, and segment EF is congruent to segment GH, then segment CD is congruent to segment GH; and if segment IJ is congruent to segment KL, then segment KL is congruent to segment IJ. The same holds true for angles. (4) two triangles are congruent if, and only if, they have 3 pairs of congruent sides and three pairs of congruent angles. In other words, triangle ABC and triangle XYZ are congruent if, and only if, angle A is congruent to angle X, angle B is congruent to angle Y, angle C is congruent to angle Z, side XY is congruent to side AB, side BC is congruent to side YZ, and side AC is congruent to side XZ. Note that when you say two triangles are congruent, you have to get the letters in the proper order to show what's congruent to what. If I had said triangle BAC was congruent to triangle YXZ, then that's okay (check it out). But if I say that triangle YZX is congruent to triangle CAB, then that's wrong. (Why?) A * X * B *--------------* C Y *------------------*Z (I'm not going to draw the diagonal lines. You can fill them in yourself.) Note that there is a LOT of things that you have to check in that last one. Six pairs of things have to be congruent. Wow. Most of the time you don't have the luxury of having all that information. Are there any shortcuts? It turns out that yes, there are some shortcuts, and those are the SSS, SAS, and ASA postulates that you mentioned. SSS: the letters stand for "Side-Side-Side". What that means is, if you have two triangles, and you can show that the three pairs of corresponding sides are congruent, then the two triangles are congruent. This is a postulate, not a theorem, meaning that it cannot be proved, but it appears to be true so everybody accepts it. So if you have triangle ANT and FLY, and you can show that AN = FL, NT = LY, and AT = FY, then the two triangles are congruent. SAS: the letters stand for "Side-Angle-Side". What that means is, if you have two triangles, call them BOY and GRL, and you can find two pairs of congruent corresponding sides AND a pair of congruent corresponding angles that is BETWEEN the two pairs of sides we just mentioned, then the two triangles are congruent. Again, this is a postulate. Suppose in triangles BOY and GRL we know that BO=RL, BY=GR, and <B=<R. I claim that triangle BOY is congruent to triangle RLG, using the SAS postulate. ASA: the letters stand for "Angle-Side-Angle". What that means is, if you have two triangles, call them DOG and CAT, and you can find two pairs of congruent corresponding angles AND a pair of congruent corresponding sides that is BETWEEN the two pairs of angles we just mentioned, then the two triangles are congruent. Again, this is a postulate. Suppose in triangles DOG and CAT we know that <D = <T, <O = <C, and DO = CT. I claim that triangle DOG is congruent to triangle TCA, using the ASA postulate. You may be wondering if there is a SSA postulate. No, unfortunately, the SSA (or ASS) postulate doesn't always work, especially in acute triangles. There is no AAA congruence postulate either, since the two triangles would have the same general shape (i.e. be similar) but one might be much bigger than the other. There is an AAS theorem, but let's not worry about that right now. If the sides and angles do not correspond, there is no congruence. For example: if we have triangle NBC and triangle KLM, and <N = <K, <B = <L, and KL = CN, then those triangles are NOT necessarily congruent. Here are four exercises for you to try. I suggest you draw a picture. (The answers are down below. Hide them from yourself.) 1. In triangle RUN and triangle HID, <R = <D, <U = <I, and RU = DI. What triangles are congruent, if any, and why? 2. In triangle FRE and SLV, FR = LV, EF = SL, and <F = <S. What triangles are congruent, if any, and why? 3. In triangle MUS and CHR, <S = <H, US = HR, and <U = <R. What triangles are congruent, if any, and why? 4. In triangle QWE and RTY, QW = TY, WE = RY, and QE = RT. What triangles are congruent, if any, and why? I'm leaving some blank space here. here's some more. Here are the answers: 4. (Yes; triangle QWE = triangle TYR by SSS) 3. (Yes; triangle SUM = triangle HRC by ASA) 2. (No; the sides and angles do not match) 1. (Yes; triangle RUN = triangle DIH by ASA) I hope this helps a little. Sorry it's so hard to draw pictures on here. -Doctor Guy, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 12/14/97 at 19:00:24 From: Franklin J. Soares Subject: Thank You!! Hi! I wrote you a question the other day and you answered it immediately. I am impressed! Your information was very very helpful. You explained everything clearly and completely. I like your method of explaining things, the way you give a definition and explain it in a complete manner. Thank you very very very much. Keep on doing what you do, it is really worth it. - Franklin Soares
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