Euclidean Geometry and its Subgeometries by Edward John Specht Harold Trainer Jones Keith G. Calkins & Donald H. Rhoads

Author:Edward John Specht, Harold Trainer Jones, Keith G. Calkins & Donald H. Rhoads
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Theorem ROT.21.

Let be a neutral plane, O be a point on , and α and β be rotations of about O. Then β ∘α = α ∘β.

Proof.

Let be any line on through O, then by Theorem ROT.13 there exist unique lines and on through O such that and , so that . Using Theorem ROT.12 and Remark NEUT.1.3 we get

. □

The following theorem shows that a rotation is what we think of as a “rigid motion.”

Theorem ROT.22.

Let O, A, and B be points in the neutral plane , where A ≠ O and B ≠ O, and let α be a rotation about O. Then .

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