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?

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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**

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