Problems on similar triangles are presented along with detailed solutions. Problem 3: The two triangles are similar and the ratio of the lengths of their sides is. The sides of two similar triangles do not have to be equal. However there is an important relationship among the sides of similar triangles: corresponding sides. Two triangles are said to be similar if their corresponding angles are congruent and the corresponding In other words, similar triangles are the same shape, but not necessarily the same size. This common ratio is called the scale factor.
The other way to think about similarity is that all of the corresponding angles will be equal.
Similar Triangle Problems
So if something is similar, then all of the corresponding angles are going to be congruent. I always have trouble spelling this. Corresponding angles are congruent. So if you have two triangles, all of their angles are the same, then you could say that they're similar.
Or if you find two triangles and you're told that they are similar triangles, then you know that all of their corresponding angles are the same. And the last way to think about it is that the sides are all just scaled-up versions of each other.
So the sides scaled by the same factor. In the example we did here, the scaling factor was 3.
It doesn't have to be 3. It just has to be the same scaling factor for every side. If we scaled this side up by 3 and we only scaled this side up by 2, then we would not be dealing with a similar triangle.
But if we scaled all of these sides up by 7, then that's still a similar, as long as you have all of them scaled up or scaled down by the exact same factor.
Similar Triangle Problems
So one way to think about it is-- I want to still visualize those triangles. Let me redraw them right over here a little bit simpler. Because I'm not talking in now in general terms, not even for that specific case.
I just redrew them so I can refer them when we write down here. If we're saying that these two things right over here are similar, that means that corresponding sides are scaled-up versions of each other.
So we could say that the length of AB is equal to some scaling factor-- and this thing could be less than some scaling factor times the length of XY, the corresponding sides.
So some scaling factor times XY. We know that the length of BC needs to be that same scaling factor times the length of YZ.
And then we know the length of AC is going to be equal to that same scaling factor times XZ. So that's XZ, and this could be a scaling factor. If they're the exact same size, if they're essentially congruent triangles, then these k's will be 1.
But another way to write these same statements-- notice, all I'm saying is corresponding sides are scaled-up versions of each other. And then the second statement right over here, if you divide both sides by YZ-- let me do it in that same color-- you get BC divided by YZ is equal to that scaling factor.
And remember, in the example we just showed, that scaling factor was 3. But now we're saying in the more general terms, similarity, as long as you have the same scaling factor. Or another way to think about it is the ratio between corresponding sides. Notice, this is the ratio between AB and XY.
So if you have similar triangles-- let me draw an arrow right over here. Similar triangles means that they're scaled-up versions, and you can also flip and rotate and do all the stuff with congruency. The statement that the point F satisfying this condition exists is Wallis's postulate  and is logically equivalent to Euclid's parallel postulate.
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: Similar triangles also provide the foundations for right triangle trigonometry. Given any two similar polygons, corresponding sides taken in the same sequence even if clockwise for one polygon and counterclockwise for the other are proportional and corresponding angles taken in the same sequence are equal in measure.
However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles otherwise, for example, all rhombi would be similar. Likewise, equality of all angles in sequence is not sufficient to guarantee similarity otherwise all rectangles would be similar. A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.
For given n, all regular n-gons are similar. Similar curves[ edit ] Several types of curves have the property that all examples of that type are similar to each other.
- Intro to triangle similarity
- Solving similar triangles