Solution to Exercise 3.45.
Be neither acute nor obtuse; aim at being just right.
—Author Unknown

Exercise 3.45. Prove Proposition 3.12.
Proposition 3.12. For an affine indirect isometry of the Euclidean plane, the measure of the angle between the two image lines has the opposite sign of the measure of the angle between the two lines.

Proof. Let the lines p' and q' be the images of lines p and q under an affine indirect isometry with matrix A. Let B be the inverse of the matrix A. By Exercise 3.44, B is the matrix of an affine indirect isometry. By Proposition 3.6, there are nonzero real numbers k₁ and k₂ such that k₁p' = pB and k₂q' = qB. We use the results of the previous two sentences together with Proposition 3.7 to compute

 

The line q' may be expressed in a similar form. Compute the measure of the angle between p' and q'.