Question

which rotation of the plane can we use to prove angles u and v are congruent, and why?
choose 1 answer:
a a 180° rotation about o maps ray \\(\overrightarrow{om}\\) onto \\(\overrightarrow{op}\\) and vice versa, and the same for rays \\(\overrightarrow{on}\\) and \\(\overrightarrow{oq}\\). this means angle u will be mapped onto angle v and vice versa.
b a rotation by the measure of angle u about o maps ray \\(\overrightarrow{om}\\) onto \\(\overrightarrow{op}\\) and vice versa, and the same for rays \\(\overrightarrow{on}\\) and \\(\overrightarrow{oq}\\). this means angle u will be mapped onto angle v and vice versa.
c a rotation by the measure of angle w about o maps ray \\(\overrightarrow{om}\\) onto \\(\overrightarrow{op}\\) and vice versa, and the same for rays \\(\overrightarrow{on}\\) and \\(\overrightarrow{oq}\\). this means angle u will be mapped onto angle v and vice versa.

Explanation:

To determine the correct rotation, we analyze the properties of angle congruence via rotations. A \(180^\circ\) rotation about a point \(O\) maps a ray to its opposite ray (since a \(180^\circ\) rotation reverses the direction of a ray through \(O\)). In the diagram, \(\overrightarrow{OM}\) and \(\overrightarrow{OP}\) are opposite rays (as are \(\overrightarrow{ON}\) and \(\overrightarrow{OQ}\)), so a \(180^\circ\) rotation about \(O\) maps \(\overrightarrow{OM}\) to \(\overrightarrow{OP}\) and \(\overrightarrow{ON}\) to \(\overrightarrow{OQ}\). This maps angle \(u\) to angle \(v\) (and vice versa), proving they are congruent.

Option B is incorrect because rotating by the measure of angle \(u\) does not necessarily map \(\overrightarrow{OM}\) to \(\overrightarrow{OP}\) (as \(u\) is not necessarily \(180^\circ\)). Option C is incorrect because rotating by the measure of angle \(w\) (which is a reflex angle or unrelated to \(180^\circ\) for opposite rays) does not map the rays appropriately. Only a \(180^\circ\) rotation about \(O\) aligns the opposite rays, mapping angle \(u\) to \(v\).

Answer:

A. A \(180^\circ\) rotation about \(O\) maps ray \(\overrightarrow{OM}\) onto \(\overrightarrow{OP}\) and vice versa, and the same for rays \(\overrightarrow{ON}\) and \(\overrightarrow{OQ}\). This means angle \(u\) will be mapped onto angle \(v\) and vice versa.