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Similar Triangles

Remember those fun diagrams from high school geometry? This post is a refresher on similar triangles, which will be needed for a forthcoming post on billiard geometry, so stay tuned for that one...

Congruent Triangles

Two triangles (or any other shape for that matter) are called congruent if they are the same shape and size, except possibly rotated. Congruent triangles thus have three sides of the same length and three angles with the same measure. If you can prove that two triangles are congruent using a few of the sides and angles, then you know the other sides and angles are also equal.

How do you prove two triangles are congruent? It turns out any of the following will work:
- side side side (SSS)
- side angle side (SAS)
- angle side angle (ASA)
- angle angle side (AAS)
- hypotenuse and one leg length (for right triangles only)

So for example, if I have two triangles with side lengths 4 and 7 with a 30 degree angle in between, then they are congruent by SAS, and the other side and two angles are also equal.

Why do these formulas work? Because given, for example, that two sides and the angle in between are all equal, then the other pieces are predetermined. That means that a second triangle with the same SAS will have the same predetermined other pieces as well, i.e. is congruent to the first triangle.

To prove that SAS works, take some triangle $ABC$ in the plane, and assume without loss of generality that point $A$ is at $(0,0)$, $B$ is on the positive $x$-axis, and $C$ is somewhere above the $x$-axis (i.e. in the positive $y$ region). This is indeed without loss of generality, because if it isn't the case, we can move our triangle left/right/up/down and/or rotate it, all without changing its shape and size.

Given the length of segment $AB$ is $L_1$, we know that $B = (L_1,0)$. Given that the measure of angle $BAC$ is $\theta$ and the length of segment $AC$ is $L_2$, we know that $C = (L_2 \cos \theta , L_2 \sin \theta)$ (and that $\theta$ will be between 0 and 180 degrees, or 0 and $\pi$ in radians, because of the assumption that $C$ is above the $x$-axis). So we've determined all 3 vertices, and thus we know all the side lengths and angles.

There are other ways to prove these as well besides coordinate proofs like the one above, but I won't go further into it since it's not that exciting, and we can at least say we proved one of them now.

Note that the following are not sufficient to establish congruence:
- AAA (proves they are the same shape, but not necessarily same size)
- ASS (which doesn't even necessarily prove they are the same shape- can you see why?)

Similar Triangles

Two triangles (or any other shape) are called similar if they are the same shape, just different size, so one is a scaled up/down version of the other. This means that the three angles are the same, and the side lengths of the second triangle are $r$ times the side lengths of the first where $r>0$ is some fixed ratio. Congruence is the special case of similarity where $r=1$.

The same five criteria I mentioned for congruence also work to prove similarity, except that for the side comparisons, you need to show that they have a certain fixed ratio instead of being equal. Since the ratio can be any positive number, AAS and ASA just become AA; if any two angles are the same, the triangles are similar. We know the third angle is the same too because the three of them need to add up to 180, so AA is equivalent to AAA and is sufficient to prove similarity. For SAS and SSS, which have more than one side involved, you need to actually check the ratio of the sides.

One last comment here: to prove that the angles add up to 180, draw a straight line $L$ through one of the vertices which is parallel to the side of the triangle which is opposite that vertex. Then the angles between the triangle and $L$ are equal to the angles at the other two vertices. But the three angles at our first vertex obviously add up to 180 because they are on a straight line.

Brief off-topic foray:

Interestingly, the angles of a triangle do not necessarily add up to 180 if the triangle is drawn on a non-flat surface (in particular, if a neighborhood of a vertex isn't flat). For example, if you draw a triangle on the sphere of the earth which has a vertex at the north pole and two vertices on the equator, you can have all 3 angles be 90 degrees, in which case the angles add up to 270.

The analog of "straight lines" on a non-flat surface, which form the sides of a triangle, are the paths of shortest distance and are called geodesics. On a sphere, the geodesics are the great circle arcs, which are pieces of circles on on the surface whose radius is the entire radius of the sphere. The equator is a great circle on the surface of the earth, as is the line dividing the eastern and western hemispheres.

Thanks for reading. Please post any questions in the comments section.


  1. Cool Post..
    Of all the posts in gtMath, I liketh this one the most
    & probably the next one too - Pool Geometry! Billiards, way to go:)

    //- ASS (which doesn't even necessarily prove they are the same shape- can you see why?)//

    Professionally, ASS never works:)
    Because, given two sides and a non-included angle, it is possible to draw 2 different triangles that satisfy those values.

    If u draw a line through the vertex, opposite to that given angle bisecting that vertex, and touching the base.. then u will end up with same length as the other side.
    So, Angle, Side, Side = ASS Does not work!

    1. Apart from sides and angles, are other properties of congruent triangles, the same too? (like area, perimeter, base, height, concentric circles etc?)
    2. Is congruence, only applicable for triangles? Can there be congruence on rectangles, quadrilaterals, polygons etc?

  2. //angles of a triangle do not necessarily add up to 180 if the triangle is drawn on earth//

    Liked this last, off-topic foray!
    U should do it for every post:)

    What about those triangles caused by umbra & penumbra of moon's shadows?
    Are they congruent or similar?:)

  3. Congruent triangles share ALL properties because they are the same triangle, just moved over or rotated. And congruence does apply to any other shape as well.

    Umbra, penumbra- I have no idea...