If a reflection in a line k followed by a reflection in a line m maps a point A to A" and this mapping is the same as a 120 degrees rotation of A about point P, what is the angle X between k and m?
When a point A is reflected in a line k, the angle between AP (where P is a point on k) and k is equal to the angle between A'P and k, where A' is the image of A after reflecting in k.
Write angle `APk = A'Pk = theta_1`
Similarly, `A'Pm = A''Pm = theta_2`
We know that A'' is the image of A after a 120 degree rotation about P. Therefore we have that
`2theta_1 + 2theta_2 = 120^o`
`implies` `theta_1 + theta_2 = 60^o`
We also know that the angle X is equal to `theta_1 + theta_2`. Therefore,
X = 60 degrees