Congruent and Similar Triangle Theorems
Date: 05/14/98 at 13:07:12 From: Amanda Crane Subject: Request 11 grade levels Dear Dr. Math, I am currently in a statistics class, but my true math love will always be geometry. So this is the subject I will ask a question about. When there are so many theorems about triangle congruency, like Angle-Side-Angle (ASA) and Side-Side-Side (SSS), why doesn't Angle- Angle-Angle (AAA) work? I will earnestly be awaiting your reply. Amanda Crane
Date: 05/16/98 at 09:25:15 From: Doctor Bob Subject: Re: Request 11 grade levels Hello Amanda, Geometry is fascinating! I hope that you will keep studying it. There are some amazing things to be learned. Now, about your question. There _is_ an Angle-Angle-Angle theorem. It just has a different conclusion than those other theorems. Remember that congruent triangles are ones in which all pairs of corresponding sides and angles have the same measurements. That means you can place one triangle on top of any congruent triangle so that all the parts coincide. The Angle-Side-Angle and Side-Side-Side theorems conclude with: "...then the two triangles are congruent." The Angle-Angle-Angle theorem I am talking about concludes with "... then the two triangles are similar." Triangles are similar if they have the same shape, but not necessarily the same size. That is, they might have congruent angles (as paired up), but the paired sides might not be congruent. To see this, take a small square cake and cut it diagonally from one corner to the opposite corner. Then throw away one half and look at the triangle remaining. (Don't do this with real cakes, your mother won't like it!) Now take a much larger square cake and cut it the same way, discard half, and look at that triangle. Those two remaining triangle cakes have the same shape but not the same size. They are "similar" but not congruent because the pairs of sides are not congruent. It is interesting that there is a Side-Angle-Side (SAS) theorem that concludes with "...then the two triangles are congruent" but there is not such a simple Side-Side-Angle (SSA) theorem, the case when the matching angles are not between the matching sides. See if you can draw a picture to show why. There are other kinds of geometry than the one you are studying. If you draw your figures on the surface of a sphere, such as a basketball, then there is an Angle-Angle-Angle theorem which gives congruent triangles. You may have to think about that for a while! Enjoy! -Doctor Bob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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