Lines l and m have an intersection at point O and any reflection in l followed by a reflection in m results in a rotation about point O of 136 degrees. What is the acute angle between l and m?
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Draw any two lines l and m to intersect at P
Pick any point P, not on either line
Then let P1 be the reflection of P in line l, and P2 the reflection of P1 in m
We now have two isosceles triangles, OP2P1 and OPP1
Also m will be a right - bisector of P1P2 at S and l will be a right - bisector of PP1 at T
l and m will also be the angle bisectors in those two isosceles triangles, thus
angle(P2OP) = 2 x angle(SOT)
136 = 2 x angle (SOT)
angle (SOT) = 68°
To find the acute angle between lines l and m, we need to determine the angle formed by the intersection point O and any point on the line l or m.
We know that a reflection in line l followed by a reflection in line m results in a rotation about point O of 136 degrees. This means that the image of a point A on line l, after reflecting it in line l and then reflecting the result in line m, will result in a point B that is rotated by 136 degrees about point O.
Let's consider a point A on line l and its image B after the reflection in line l and then reflection in line m. The resulting angle between OA and OB is 136 degrees.
Since the rotation angle about point O is 136 degrees, we can deduce that the angle AOB is twice the angle between lines l and m.
Therefore, the acute angle between lines l and m is 136 degrees divided by 2, which equals 68 degrees.