Triangle Congruence - SIDE SIDE SIDE - Proof

Описание: In this proof I look at why two triangles with the same edges, a, b and c must be congruent.

I begin by showing that there are only two possible places for the left and right edges of a triangle can meet at the same point.

Connecting the left and right edges at one of these points, I then show that any change in either of the base angles would break the triangle. So there is only one valid angle between base and left edge, and another valid angle between base and right edge in order for the triangle to exist; it is impossible to have an a, b, c triangle with angle values other than those two valid angles.

That also means that, since the third angle depends on the size of the other two, there can only be one valid value for the third angle.

This already proves that all triangles with edges of length a, b and c must be copies of the only possible triangle. But I go on to create a triangle using the other possible meeting points of the right and left edges, which makes another triangle possible.

The second triangle is then examined, and manipulating the triangles so that, together, they form a parallelogram, the properties of a parallelogram are highlighted and used to show that the second triangle is identical to the first.

The above shows that although there are two point where the edges can meet in order to create a triangle, both of the triangles created, one at each point, are identical. Hence we've proved that multiple triangles that have exactly the same edges are congruent.

The below linked video is a good refresher for parallelogram properties, and will help you understand the second part of this proof better; check it out:
   • Angle Properties of a PARALLELOGRAM - How ...  



Chapters:
0:00 intro and reason for video
0:40 building the proof structure
1:24 left and right edges can be positioned in many places
2:25 there are only two points in space where a triangle can be created
3:03 creating a triangle by joining left and right edges at one of the two points
3:29 changing either base angle breaks the triangle, so triangle a, b, c MUST have precise angles related to lengths a, b and c.
3:52 creating and examining a second triangle, created by the second point.
4:49 proving the triangles are identical
7:57 conclusion: two triangles with edges of lengths a, b and c, must be identical.

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