Theorem 154

This worksheet is a visualization of the proof of Theorem 154. This theorem states that any composition of two rotations whose angles add up to be 0 degrees mod 360 is in fact a translation. In the worksheet, both the composition of rotations, blue, and composition of reflections, brown, are shown. You can see that the blue and brown actually overlap on two of the triangles. Since rotations are a composition of two reflections, the paths that the rotation and reflections take will intersect at two places. Another thing to notice is that the triangle after the first reflection is also the same triangle after the third reflection. When each rotation is composed as reflections, they share a reflection line, line f. Since reflecting over the same line twice is an identity, reflecting over line f twice does not move the triangle. Because we have an identity with the reflection over line f twice, we are essentially reflecting over two parallel lines. This we know is a translation and is shown here. So in short, because a rotation is a composition of two reflections, these two rotations share the same reflection line and reflecting over two parallel lines results in a translation, you can see that two rotations whose angles add up to 0 degrees mod 360 is a translation.