Question

1. use the diagram and the given information to find the measures of the angles in parts (a)-(d).

. \\( \overrightarrow{a b} \\) and \\( \overrightarrow{c d} \\) are cut by the transversal \\( \overrightarrow{e f} \\).
. assume \\( \overrightarrow{a b} \\) is parallel to \\( \overrightarrow{c d} \\).
. the measure of \\( \angle e g b \\) is \\( 60^{circ} \\).
a. \\( \angle a g h \\) b. \\( \angle c h f \\) c. \\( \angle g h d \\) d. \\( \angle e g a \\)

Explanation:

Step1: Find \(\angle AGH\)

Since \(\angle EGB = 60^{\circ}\) and \(\angle AGH\) and \(\angle EGB\) are vertical angles. Vertical angles are equal.
\(\angle AGH=\angle EGB = 60^{\circ}\)

Step2: Find \(\angle CHF\)

Because \(\overrightarrow{AB}\parallel\overrightarrow{CD}\), \(\angle EGB\) and \(\angle CHF\) are corresponding angles. Corresponding angles are equal when two parallel lines are cut by a transversal.
\(\angle CHF=\angle EGB = 60^{\circ}\)

Step3: Find \(\angle GHD\)

Since \(\overrightarrow{AB}\parallel\overrightarrow{CD}\), \(\angle EGB\) and \(\angle GHD\) are alternate - interior angles. Alternate - interior angles are equal when two parallel lines are cut by a transversal.
\(\angle GHD=\angle EGB = 60^{\circ}\)

Step4: Find \(\angle EGA\)

\(\angle EGA\) and \(\angle EGB\) are supplementary angles (they form a linear pair). The sum of supplementary angles is \(180^{\circ}\).
\(\angle EGA=180^{\circ}-\angle EGB\)
Substitute \(\angle EGB = 60^{\circ}\) into the formula:
\(\angle EGA=180^{\circ}- 60^{\circ}=120^{\circ}\)

Answer:

a. \(60^{\circ}\)
b. \(60^{\circ}\)
c. \(60^{\circ}\)
d. \(120^{\circ}\)