Question

complete the diagram using the given information.
l and m are parallel

1. draw a □ at ∠b and its corresponding angle.
2. draw a ○ at all angles with the same measure as ∠g. what is the measure of these angles?
3. draw a △ at the angle that is vertical to ∠a.
4. ∠a and ∠f are

a) alternate exterior
b) alternate interior
c) same side exterior
d) same side interior

1. ∠f and ∠g are (choose any that apply)

a) alternate exterior
b) same side exterior
c) supplementary
d) corresponding to ∠b and ∠c

Explanation:

Step1: Find the measure of \(\angle C\)

Since \(\angle A = 42^{\circ}\), and \(\angle A\) and \(\angle C\) are supplementary (they form a linear pair), we have \(\angle A+\angle C = 180^{\circ}\). So, \(42^{\circ}+\angle C=180^{\circ}\), then \(\angle C = 180^{\circ}- 42^{\circ}=138^{\circ}\).

Step2: Use the property of parallel lines

Because \(L\parallel M\), corresponding angles are equal. Angles with the same measure as \(\angle C\) (using properties of parallel lines: corresponding angles, alternate - interior angles, vertical angles) will have a measure of \(138^{\circ}\).

Step3: Analyze \(\angle A\) and \(\angle F\)

\(\angle A\) and \(\angle F\) are alternate exterior angles. Alternate exterior angles are formed when a transversal crosses two parallel lines. The transversal \(T\) crosses parallel lines \(L\) and \(M\). \(\angle A\) is outside line \(L\) and \(\angle F\) is outside line \(M\) and on opposite sides of the transversal.

Step4: Analyze \(\angle F\) and \(\angle G\)

\(\angle F\) and \(\angle G\) form a linear pair (they are adjacent and their non - common sides form a straight line), so \(\angle F+\angle G=180^{\circ}\), which means they are supplementary. Also, \(\angle F\) corresponds to \(\angle B\) and \(\angle G\) corresponds to \(\angle C\) (by the property of parallel lines and transversal, corresponding angles are in the same relative position with respect to the parallel lines and the transversal)

Answer:

1. (Diagrammatic step, not shown in text here but conceptually: \(\angle B\) and its corresponding angle (say \(\angle E\)) are in the same relative position with respect to the parallel lines \(L\) and \(M\) and the transversal)
2. \(138^{\circ}\)
3. (Diagrammatic step, the vertical angle to \(\angle A\) is the angle opposite to \(\angle A\) at the intersection of line \(T\) and line \(L\))
4. a) alternate exterior
5. c) supplementary, d) corresponding to \(\angle B\) and \(\angle C\)