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But now, instead of connecting G to A, we'll draw the angle bisector of ∠GBA, and extend it until it intersects CG at point H: |GC|>|AC|, and since |GC|=|DF|, |DF|>|AC| Second method - using the triangle inequalityįor the second method of proving the Hinge Theorem, we'll construct the same new triangle, △GBC, as before. So ∠CAG>∠BAG=∠BGA>∠CGA, and so ∠CAG>∠CGA.Īnd now, from the converse of the scalene triangle Inequality, the side opposite the large angle (GC) is larger than the one opposite the smaller angle (AC). From the angle addition postulate, ∠BGA>∠CGA, and also ∠CAG>∠BAG. From the Base Angles theorem, we have ∠BGA= ∠BAG. |GB|=|AB| by construction, so △GBA is isosceles. This triangle has side AC, and from the above congruent triangles, side |GC|=|DF|. To put the edges that we want to compare in a single triangle, we'll draw a line from G to A. Let's look at the first method for proving the Hinge Theorem. First method - using the converse scalene triangle inequality So the two triangles are congruent by the Side-Angle-Side postulate, and as a result |GC|=|DF|. We have |DE=|GB| by construction, θ 2=∠DEF=∠GBC by construction, and |BC|=|EF| (given). We'll now compare the newly constructed triangle △GBC to △DEF. In practice, we will use a compass and straight edge to construct a new triangle, △GBC, by copying angle θ 2 into a new angle ∠GBC, and copying the length of DE onto the ray BG so that |DE=|GB|=|AB|. The above description was a colloquial, layman's description of what we are doing. Let's place triangle △ABC over △DEF so that one of the congruent edges overlaps, and since θ 2>θ 1, the other congruent edge will be outside △ABC: So the first order of business is to get these sides into one triangle. We'll use each one of these in the two different ways we prove the Theorem.īut one hurdle first: both these theorems deal with sides (or angles) of a single triangle. The other is the triangle inequality theorem, which tells us the sum of any two sides of a triangle is larger than the third side. This tells us that the side facing the larger angle is larger than the side facing the smaller angle. One of these is the converse of the scalene triangle Inequality. This guides us to use one of the triangle inequalities which provide a relationship between sides of a triangle. To prove the Hinge Theorem, we need to show that one line segment is larger than another. Two triangles, △ABC and △DEF, have two pairs of congruent sides: |AB|=|DE| |BC|=|EF|. It is also sometimes called the "Alligator Theorem" because you can think of the sides as the (fixed length) jaws of an alligator- the wider it opens its mouth, the bigger the prey it can fit. The Hinge Theorem states that in the triangle where the included angle is larger, the side opposite this angle will be larger. Think of it as a hinge, with fixed sides, that can be opened to different angles: What is the Hinge Theorem? Let's say you have a pair of triangles with two congruent sides but a different angle between those sides. |
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