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Very first method – using the converse scalene triangle inequality

Very first method – using the converse scalene triangle inequality

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What is the Depend Theorem? What if you’ve got a couple of triangles having one or two congruent edges but yet another direction anywhere between those sides. Consider it given that a beneficial rely, with fixed sides, which may be unsealed to various bases:

The new Depend Theorem states one throughout the triangle in which the included angle try huge, the side contrary that it position could well be huge.

It is quite both known as “Alligator Theorem” as you may think of the sides as (fixed duration) mouth area out of an alligator- the new broad they opens their mouth, the bigger new target it can complement.


To show new Depend Theorem, we should instead demonstrate that one-line part was bigger than another. Both traces are also corners during the an excellent triangle. It guides us to have fun with among the triangle inequalities and that render a relationship anywhere between corners out-of an effective triangle. One among them ‘s the converse of the scalene triangle Inequality.

So it tells us the front against the larger angle try bigger than along side it up against the smaller direction. The other ‘s the triangle inequality theorem, and that tells us the sum of any a couple sides of an excellent triangle try bigger than the 3rd side.

But that difficulty earliest: both these theorems manage sides (or bases) of one triangle. Right here i’ve several independent triangles. And so the first-order regarding business is to find these sides to the you to triangle.

Let’s place triangle ?ABC over ?DEF so that one of the congruent edges overlaps, and since ?2>?1, the other congruent edge will be outside ?ABC:

The above description was a colloquial, layman’s description of what we are doing. In practice, we will use a compass and straight edge to construct a new triangle, ?GBC, by copying angle ?2 into a new angle ?GBC, and copying the length of DE onto the ray BG so that |DE=|GB|=|AB|.

We’ll now compare the newly constructed triangle ?GBC to ?DEF. We have |DE=|GB| by construction, ?2=?DEF=?GBC by construction, and |BC|=|EF| (given). So the two triangles are congruent by the Side-Angle-Side postulate, and as a result |GC|=|DF|.

Why don’t we look at the earliest method for indicating the new Rely Theorem. To get the latest edges that we want to contrast from inside the an effective single triangle, we are going to mark a line out-of Grams to An effective. So it forms a different triangle, ?GAC. So it triangle enjoys side Air-con, and you can regarding a lot more than congruent triangles, top |GC|=|DF|.

Today let’s consider ?GBA. |GB|=|AB| because of the build, so ?GBA try isosceles. Regarding the Ft Bases theorem, we have ?BGA= ?Handbag. On the angle introduction postulate, ?BGA>?CGA, and have ?CAG>?Bag. So ?CAG>?BAG=?BGA>?CGA, thereby ?CAG>?CGA.

And now, from the converse of the scalene triangle Inequality, the medial side opposite the massive perspective (GC) try larger than the main one contrary the smaller perspective (AC). |GC|>|AC|, and since |GC|=|DF|, |DF|>|AC|

Next means – utilizing the triangle inequality

Towards the 2nd form of demonstrating the latest Count Theorem, we shall make a comparable the new triangle, ?GBC, as the just before. However now, instead of connecting G in order to An excellent, we’ll mark the latest angle bisector off ?GBA, and you will increase it until it intersects CG at section H:

Triangles ?BHG and you may ?BHA is congruent by the Side-Angle-Front postulate: AH is a common front side, |GB|=|AB| because of the framework and ?HBG??HBA, since BH is the position bisector. Because of this |GH|=|HA| as associated sides inside the congruent triangles.

Today believe triangle ?AHC. About triangle inequality theorem, we have tagged nedir |CH|+|HA|>|AC|. But since |GH|=|HA|, we can alternative and have now |CH|+|GH|>|AC|. But |CH|+|GH| is largely |CG|, therefore |CG|>|AC|, so when |GC|=|DF|, we have |DF|>|AC|

And thus we had been in a position to confirm the fresh Count Theorem (known as brand new Alligator theorem) in two implies, depending on the fresh new triangle inequality theorem otherwise their converse.

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