Very first approach – utilizing the converse scalene triangle inequality

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What’s the Depend Theorem? What if you have got a set of triangles having several congruent corners but another position anywhere between those edges. Think of it since the an excellent rely, having repaired sides, which are exposed to various angles:

Brand new Depend Theorem claims you to definitely on triangle where in actuality the integrated direction was large, the medial side contrary that it position was big.

It’s very possibly known as “Alligator Theorem” because you can think of the corners due to the fact (repaired length) jaws out of a keen alligator- the large they opens their mouth, the larger brand new target it does complement.

Strategy

To prove the fresh Depend Theorem, we must reveal that one line phase try larger than other. Both traces are also edges into the an excellent triangle. It courses us to explore one of many triangle inequalities and that give a relationship anywhere between corners out-of an effective triangle. One is the converse of your scalene triangle Inequality.

So it tells us the front against the bigger direction was bigger than along side it against small position. Additional is the triangle inequality theorem, and that informs us the sum any several corners out of an effective triangle is bigger than the third side.

However, you to definitely hurdle basic: both of these theorems handle edges (or angles) of just one triangle. Here you will find several independent triangles. So the first order from organization is discover these types of corners on 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, PouЕѕijte Weblink. 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|.

Let us glance at the basic way for proving the fresh new Hinge Theorem. To place the brand new edges that we want to contrast inside the a solitary triangle, we’ll draw a column off Grams to A beneficial. Which forms yet another triangle, ?GAC. Which triangle enjoys side Air cooling, and you may from the more than congruent triangles, front |GC|=|DF|.

Today why don’t we look at ?GBA. |GB|=|AB| from the construction, thus ?GBA was isosceles. Throughout the Ft Bases theorem, we have ?BGA= ?Wallet. Regarding angle introduction postulate, ?BGA>?CGA, and just have ?CAG>?Purse. Thus ?CAG>?BAG=?BGA>?CGA, and so ?CAG>?CGA.

And from now on, from the converse of your scalene triangle Inequality, the side reverse the large angle (GC) are larger than the one reverse the smaller direction (AC). |GC|>|AC|, and since |GC|=|DF|, |DF|>|AC|

2nd approach – making use of the triangle inequality

Into next style of showing new Depend Theorem, we will create an equivalent this new triangle, ?GBC, since the ahead of. However now, instead of hooking up Grams in order to A, we shall draw the fresh new perspective bisector from ?GBA, and you will offer they until they intersects CG in the point H:

Triangles ?BHG and you will ?BHA is actually congruent of the Front side-Angle-Front postulate: AH is a common side, |GB|=|AB| because of the build and you will ?HBG??HBA, once the BH is the position bisector. Because of this |GH|=|HA| as associated corners in congruent triangles.

Now imagine triangle ?AHC. From the triangle inequality theorem, i have |CH|+|HA|>|AC|. But since the |GH|=|HA|, we can alternative and then have |CH|+|GH|>|AC|. However, |CH|+|GH| is basically |CG|, thus |CG|>|AC|, so that as |GC|=|DF|, we obtain |DF|>|AC|

And therefore we were able to prove new Rely Theorem (called the newest Alligator theorem) in two implies, counting on the latest triangle inequality theorem otherwise the converse.