Question

peng was asked to find the measure of angle ( x ) and explain his reasoning.

image of intersecting lines with points o, h, k, i, j and angle 120° and angle ( x )

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 dropdown.
• ray dropdown maps onto ray ( overrightarrow{ok} ).
• ray ( overrightarrow{ok} ) maps onto ray ( overrightarrow{oj} ).

therefore, the image of dropdown will be exactly where the pre - image of angle ( x ) was. since rotations preserve angle measure, the measure of angle ( x ) must be ( 120^circ ).

Explanation:

Step1: Analyze Ray \( \overrightarrow{OI} \) mapping

A \( 180^\circ \) rotation about \( O \) means a ray and its image are opposite rays. Since \( \overrightarrow{OH} \) maps to \( \overrightarrow{OI} \), then \( \overrightarrow{OI} \) should map to \( \overrightarrow{OH} \) (as they are opposite rays after \( 180^\circ \) rotation).

Step2: Analyze ray mapping to \( \overrightarrow{OK} \)

For a ray to map to \( \overrightarrow{OK} \) under \( 180^\circ \) rotation, it should be the opposite ray of \( \overrightarrow{OK} \), which is \( \overrightarrow{OJ} \) (since \( \overrightarrow{OK} \) maps to \( \overrightarrow{OJ} \), so the reverse: \( \overrightarrow{OJ} \) maps to \( \overrightarrow{OK} \)).

Step3: Analyze angle image

The angle with measure \( 120^\circ \) (between \( \overrightarrow{OH} \) and \( \overrightarrow{OK} \)) will have its image as angle \( x \) (between \( \overrightarrow{OI} \) and \( \overrightarrow{OJ} \)) under \( 180^\circ \) rotation. So the image of the \( 120^\circ \) angle (between \( \overrightarrow{OH} \) and \( \overrightarrow{OK} \)) will be angle \( x \) (between \( \overrightarrow{OI} \) and \( \overrightarrow{OJ} \)).

Answer:

• Ray \( \overrightarrow{OI} \) maps onto \( \boldsymbol{\overrightarrow{OH}} \).
• Ray \( \boldsymbol{\overrightarrow{OJ}} \) maps onto ray \( \overrightarrow{OK} \).
• Therefore, the image of \( \boldsymbol{\text{the } 120^\circ \text{ angle (between } \overrightarrow{OH} \text{ and } \overrightarrow{OK})} \) will be exactly where the pre - image of angle \( x \) was. And the measure of angle \( x \) is \( \boldsymbol{120^\circ} \).

(For the blanks in order: \( \overrightarrow{OH} \), \( \overrightarrow{OJ} \), the \( 120^\circ \) angle (between \( \overrightarrow{OH} \) and \( \overrightarrow{OK} \)))