### First method – making use of the converse scalene triangle inequality

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What’s the Hinge Theorem? Imagine if you’ve got a pair of triangles with a couple congruent sides but another perspective ranging from those individuals edges. View it while the a depend, which have repaired sides, that may be unsealed to several basics:

The fresh Count Theorem says one to on the triangle where in actuality the incorporated angle try larger, the medial side reverse this angle is big.

It can be sometimes known as “Alligator Theorem” because you can think about the corners since the (repaired length) oral cavity regarding an alligator- the new wide it opens up the lips, the greater the newest prey it will complement.

To show the new Hinge Theorem, we must demonstrate that one-line sector is larger than some other. Both outlines also are corners inside a beneficial triangle. That it books us to use one of several triangle inequalities hence offer a love between sides of an excellent triangle. One of these ‘s the converse of the scalene triangle Inequality.

This informs us that front up against the bigger perspective is actually bigger than the medial side against small angle. One other is the triangle inequality theorem, and therefore tells us the sum of the one several corners out of a triangle was bigger than the 3rd front side.

But you to hurdle very first: both these theorems deal with edges (otherwise basics) of step one triangle. Here i’ve two independent triangles. Therefore, the first-order out-of company is to track down these corners into that 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|.

Let’s glance at the first method for showing the Rely Theorem. To get the fresh edges that individuals want to evaluate for the a single triangle, we will mark a line from G so you can A good. This forms another triangle, ?GAC. So it triangle enjoys front Air-con, and you can throughout the over congruent triangles, front |GC|=|DF|.

Now let us check ?GBA. |GB|=|AB| from the structure, thus twoo Seznamka?GBA was isosceles. On Foot Basics theorem, we have ?BGA= ?Handbag. On angle addition postulate, ?BGA>?CGA, while having ?CAG>?Wallet. Thus ?CAG>?BAG=?BGA>?CGA, and therefore ?CAG>?CGA.

Now, throughout the converse of scalene triangle Inequality, the side opposite the enormous angle (GC) try larger than usually the one contrary the smaller direction (AC). |GC|>|AC|, and since |GC|=|DF|, |DF|>|AC|

## Second method – using the triangle inequality

To your next type exhibiting the fresh Count Theorem, we’ll construct the same new triangle, ?GBC, since before. But now, unlike linking G to help you An effective, we’ll mark the fresh new perspective bisector off ?GBA, and stretch they up until it intersects CG within part H:

Triangles ?BHG and you will ?BHA try congruent by Side-Angle-Front postulate: AH is a common top, |GB|=|AB| from the structure and you will ?HBG??HBA, due to the fact BH is the perspective bisector. As a result |GH|=|HA| as associated corners in the congruent triangles.

Today thought triangle ?AHC. Regarding the triangle inequality theorem, i’ve |CH|+|HA|>|AC|. But because |GH|=|HA|, we are able to replace and now have |CH|+|GH|>|AC|. However, |CH|+|GH| is actually |CG|, thus |CG|>|AC|, so that as |GC|=|DF|, we obtain |DF|>|AC|

And thus we had been in a position to prove new Hinge Theorem (known as the Alligator theorem) in 2 ways, counting on the new triangle inequality theorem or their converse.