Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one (Equal angles have been marked with the same number of arcs) . When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangles. In Figure 1, &De. Definition: Two triangles are said to be similar if they have the same angle is an important relationship among the sides of similar triangles: corresponding.
This is equivalent to saying that one triangle or its mirror image is an enlargement of the other. Two sides have lengths in the same ratio, and the angles included between these sides have the same measure.
This is known as the SAS similarity criterion. Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other transitivity of similarity of triangles.
Corresponding altitudes of similar triangles have the same ratio as the corresponding sides.
Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. The statement that the point F satisfying this condition exists is Wallis's postulate  and is logically equivalent to Euclid's parallel postulate.Similar Triangles
In the axiomatic treatment of Euclidean geometry given by G. Birkhoff see Birkhoff's axioms the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms.
Among the elementary results that can be proved this way are: Now the whole reason I did that is to leverage that, corresponding sides, the ratio between corresponding sides of similar triangles, is always going to be the same. These are similar triangles.
Triangle similarity & the trigonometric ratios
They're corresponding to each other. And we could keep going, but I'll just do another one.
And we got all of this from the fact that these are similar triangles. So this is true for any right triangle that has an angle theta. Then those two triangles are going to be similar, and all of these ratios are going to be the same. Well, maybe we can give names to these ratios relative to the angle theta. So from angle theta's point of view-- I'll write theta right over here, or we can just remember that-- what is the ratio of these two sides?
Well from theta's point of view, that blue side is the opposite side. It's opposite-- so the opposite side of the right triangle. And then the orange side we've already labeled the hypotenuse. So from theta's point of view, this is the opposite side over the hypotenuse. And I keep stating from theta's point of view because that wouldn't be the case for this other angle, for angle B. From angle B's point of view, this is the adjacent side over the hypotenuse.
And we'll think about that relationship later on.
But let's just all think of it from theta's point of view right over here. So from theta's point of view, what is this? Well theta's right over here.
Clearly AB and DE are still the hypotenuses-- hypoteni. I don't know how to say that in plural again. And what is AC, and what are DF? Well, these are adjacent to it.
Similar Triangles: Perimeters and Areas
They're one of the two sides that make up this angle that is not the hypotenuse. So this we can view as the ratio, in either of these triangles, between the adjacent side-- so this is relative. Once again, this is opposite angle B, but we're only thinking about angle A right here, or the angle that measures theta, or angle D right over here-- relative to angle A, AC is adjacent.
Relative to angle D, DF is adjacent. So this ratio right over here is the adjacent over the hypotenuse. And it's going to be the same for any right triangle that has an angle theta in it.
And then finally, this over here, this is going to be the opposite side. Once again, this was the opposite side over here. This ratio for either right triangle is going to be the opposite side over the adjacent side. And I really want to stress the importance-- and we're going to do many, many more examples of this to make this very concrete-- but for any right triangle that has an angle theta, the ratio between its opposite side and its hypotenuse is going to be the same. That comes out of similar triangles.
We've just explored that.
Triangle similarity & the trigonometric ratios (video) | Khan Academy
The ratio between the adjacent side to that angle that is theta and the hypotenuse is going to be the same, for any of these triangles, as long as it has that angle theta in it. And the ratio, relative to the angle theta, between the opposite side and the adjacent side, between the blue side and the green side, is always going to be the same.
So given that, mathematicians decided to give these things names. Relative to the angle theta, this ratio is always going to be the same, so the opposite over hypotenuse, they call this the sine of the angle theta.
Let me do this in a new color-- by definition-- and we're going to extend this definition in the future-- this is sine of theta. This right over here, by definition, is the cosine of theta. And this right over here, by definition, is the tangent of theta. And a mnemonic that will help you remember this-- and these really are just definitions. People realized, wow, by similar triangles, for any angle theta, this ratio is always going to be the same.
Because of similar triangles, for any angle theta, this ratio is always going to be the same.