Angles Between a Line and its Rotated Image

Rotations change the directions of lines and segments by the angle of rotation.  This page tells what this means, starting with the simple unoriented statement and moving to the oriented statement.

Angles between two lines

If m and n intersect at P, then the two lines form 2 pairs of vertical angles.  One pair of angles has the measure a degrees and the others are supplementary and have the measure 180 – a degrees.  When one speaks of the angle between two lines, there are really two supplementary angles and it may not be clear which to choose.

If m = PQ and n = PR, then the rays (halflines) PQ and PR define s single angle PQR.  This picks out one of the angles between the lines.  It also gives an oriented angle.

Rotating lines through the Center of Rotation

Suppose an isometry T = rotation Ax. 

Suppose that m = AB is a line through A.  Let B' = T(B).  By definition of rotation, the oriented angle BAB' = x.  So the angle between line m and line m' = T(m) = x. This can be taken as the oriented angle from m to its rotated image.

Rotating lines NOT  through the Center of Rotation

Suppose an isometry T = rotation Ax. 

Suppose that m is a line not through A.  Let F be the intersection of the line m with the line through A perpendicular to m (i.e., F is the foot of the perpendicular).

Denote by  F'  and m' be the images of the points and the line by T.  Since T is an isometry, the line AF' is perpendicular to m' at F'.

Line AF is rotated to line AF', with angle FAF' = x., since line AF passes through A.

If x = 180 degrees, then the points F, A, F' are collinear and the perpendicular lines m and m' are parallel.

If x is not 180 degrees, then the lines m and m' do not have a common perpendicular, so are intersecting lines.  Let the point of intersection be G.

Assume that 0 < x < 180.  Then the quadrilateral AFGF' has two right angles at F and F'.  Since the angle sum = 360 degrees, this means that the remaining angles at A and B must add up to 180 degrees.  So the angle FGF' = 180 – x  and the supplementary angle that is an exterior angle of AFF'G at G is the angle x.

If -180 <  x < 0 the figure that is drawn is the mirror image of the figure above, so the unoriented angles are the same, and the oriented angles are still equal but now are negative.