Unizor - Geometry2D - Similarity - Angles
Channel: Zor Shekhtman on UNIZOR Education
Date: 2012-05-18

Scaling transforms a straight line into a line parallel to the original one. To prove it, assume the opposite, that is the lines are not parallel. Then they intersect each other at some point X. If Q is a center of scaling, M - any point on the original line and N - its image on the transformed line, points Q, M and N lie on the same line and the ratio between lengths of segments QN and QM is a factor of scaling, the same for all points on a line. Now, we assumed that the lines intersect at point X. That means that point X is transformed into itself, which, in turn, means that the factor of scaling is equal to 1. But in this case all points of the original line will not move during this transformation and original line completely coincides with its transformed image. Therefore, we have proved that original line and its image either do not have any common points or completely coincide with each other. Therefore, scaling of a straight line transforms it into a line parallel to the original. End of proof.

Let's use this property of scaling to prove that angles are transformed into angles of equal measure (that is congruent to original angles). Consider an angle formed by two rays and a transformation of scaling with certain center and factor. The image of the first ray, according to a theorem about parallelism of original line and its image, is a ray parallel to the original one. Similarly, the image of the second ray is parallel to it. Therefore, the image of an angle is an angle with legs, correspondingly parallel to the rays forming the original angle. As we know, these two angles with correspondingly parallel legs are congruent and have the same measure. End of proof.