Question

fill in the blanks in pengs solution.
if we perform a $180^{\circ}$ rotation about $o$, the following will happen:

• ray $\overrightarrow{oh}$ maps onto ray $\overrightarrow{oi}$.
• ray $\overrightarrow{oi}$ maps onto ray $oh$
• ray ray $oj$ maps onto ray $\overrightarrow{ok}$.
• ray $\overrightarrow{ok}$ maps onto ray $\overrightarrow{oj}$.

therefore, the image of $\square$ will be exactly where the pre - image of angle $x$ was. since rotat measure, the measure of angl $x$ must be $120^{\circ}$.
$\angle iok$
$\angle hoj$
$\angle koh$
6 of 30

Explanation:

Step1: Recall Rotation Properties

A \(180^\circ\) rotation about a point maps a ray to the ray opposite to it (since \(180^\circ\) rotation reverses the direction). So, if \(\overrightarrow{OH}\) maps to \(\overrightarrow{OI}\), then \(\overrightarrow{OI}\) should map to \(\overrightarrow{OH}\) (opposite ray relationship). Also, for the ray mapping to \(\overrightarrow{OK}\), since \(\overrightarrow{OK}\) maps to \(\overrightarrow{OJ}\), the ray mapping to \(\overrightarrow{OK}\) should be \(\overrightarrow{OJ}\) (reverse of the mapping \(\overrightarrow{OK}\to\overrightarrow{OJ}\) is \(\overrightarrow{OJ}\to\overrightarrow{OK}\)).

Step2: Analyze Angle Image

For the angle, a \(180^\circ\) rotation preserves angle measure (rotation is a rigid transformation, so angle measures are invariant). The angle with measure \(120^\circ\) (given) and angle \(x\) are related by rotation. The angle whose image is \(x\) (or vice - versa) should have the same measure. From the ray mappings, the angle \(\angle HOJ\) (or the angle formed by \(\overrightarrow{OH}\) and \(\overrightarrow{OJ}\)) and the angle related to \(x\) (formed by \(\overrightarrow{OI}\) and \(\overrightarrow{OK}\) maybe) – but since rotation preserves measure, if the pre - image angle (the \(120^\circ\) angle) maps to \(x\) (or \(x\) maps to it), their measures are equal. So the angle whose image is \(x\) is \(\angle HOJ\) (or the \(120^\circ\) angle), and thus \(x = 120^\circ\).

Answer:

The measure of angle \(x\) is \(120^\circ\), and the relevant ray mappings are: Ray \(\overrightarrow{OI}\) maps onto ray \(\overrightarrow{OH}\), Ray \(\overrightarrow{OJ}\) maps onto ray \(\overrightarrow{OK}\), and the image of \(\angle HOJ\) (or the \(120^\circ\) angle) is related to angle \(x\) such that \(x = 120^\circ\).