First means – with the converse scalene triangle inequality

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What is the Rely Theorem? What if you may have a pair of triangles that have a few congruent sides but a separate direction ranging from people edges. Consider it because the a hinge, that have fixed corners, which might be open to different angles:

The newest Depend Theorem says you to throughout the triangle where in fact the provided position was big, along side it contrary this angle would-be large.

It is quite both known as “Alligator Theorem” because you can think of the corners just like the (fixed size) mouth area out-of a keen alligator- this new broad they opens the mouth, the greater the brand new victim it will complement.

Method

To show the Count Theorem, we have to show that one-line part is bigger than other. Each other outlines also are corners in the a great triangle. Which guides us to play with among the triangle inequalities and therefore provide a love ranging from sides regarding a great triangle. One among these ‘s the converse of scalene triangle Inequality.

Which confides in us the top facing the bigger angle try bigger than along side it up against the smaller direction. Additional ‘s the triangle inequality theorem, and therefore tells us the sum people a couple of edges out of an effective triangle is bigger than the third front side.

But you to definitely difficulty very first: both of these theorems manage edges (or angles) of 1 triangle. Here we have one or two independent triangles. And so the first order off business is to acquire this type of sides into 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|.

Let us look at the basic opportinity for demonstrating this new Depend Theorem. To put new corners that individuals want to compare into the a great solitary triangle, we’re going to mark a column out-of Grams in order to A great. That it versions a separate triangle, ?GAC. This triangle provides front Ac, and you will from the more than congruent triangles, front |GC|=|DF|.

Now why don’t we take a look at ?GBA. |GB|=|AB| of the structure, thus ?GBA is actually isosceles. Throughout the Foot Angles theorem, i have ?BGA= ?Wallet. Regarding the position addition postulate, ?BGA>?CGA, and now have ?CAG>?Wallet. Very ?CAG>?BAG=?BGA>?CGA, and therefore ?CAG>?CGA.

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

Next strategy – with the triangle inequality

With the next type of showing this new Rely Theorem, we will make a similar the fresh triangle, ?GBC, since the just before. However now, in lieu of connecting Grams so you’re able to Good, we’re going to mark the perspective bisector away from ?GBA, and you will extend it until it intersects CG within point H:

Triangles ?BHG and you will ?BHA try congruent of the Front-Angle-Top postulate: AH is a common side, |GB|=|AB| of the build and you may ?HBG??HBA, as BH ‘s the perspective bisector. As a result |GH|=|HA| since associated edges inside the congruent triangles.

Now envision triangle ?AHC. Regarding the triangle inequality theorem, you will find |CH|+|HA|>|AC|. However, because the |GH|=|HA|, we are able to alternative and get |CH|+|GH|>|AC|. But |CH|+|GH| is actually |CG|, therefore |CG|>|AC|, and as |GC|=|DF|, we become |DF|>|AC|

And thus we were able to confirm the latest Count Theorem (called the brand new Alligator theorem) in two ways, depending on the brand new triangle inequality theorem otherwise their converse.