Question

use the diagram of g||h, with transversal f, to answer the questions.
a pair of corresponding angles is
angle 4 and angle 8.
angle 2 is
angle 8.
a pair of sa or angles is
greater than
less than
congruent to

Explanation:

Step1: Recall Parallel Line Angles

When two parallel lines (\(g \parallel h\)) are cut by a transversal (\(f\)), corresponding angles are congruent, alternate - interior angles are congruent, and consecutive - interior angles are supplementary. Also, we can use the properties of vertical angles and linear pairs. Angle 2 and angle 8: Let's analyze the positions. Angle 2 and angle 5 are corresponding angles (since \(g\parallel h\) and \(f\) is transversal), angle 5 and angle 8 are vertical angles? No, angle 5 and angle 8 are supplementary? Wait, no. Wait, angle 2 and angle 3 are vertical? No, angle 2 and angle 4 are vertical? Wait, let's look at the lines. Line \(g\) and line \(h\) are parallel, transversal \(f\). Angle 2 and angle 8: Let's see, angle 2 and angle 5 are corresponding (so congruent), angle 5 and angle 8 are same - side interior? No, angle 5 and angle 8 are adjacent? Wait, no. Wait, angle 2 and angle 8: Let's use the property of parallel lines. Angle 2 and angle 3 are supplementary (linear pair), angle 3 and angle 8: are they alternate - interior? Wait, line \(g\) and \(h\) are parallel, transversal \(f\). Angle 3 and angle 8: angle 3 is on line \(g\), below \(f\), and angle 8 is on line \(h\), below \(f\), and they are on alternate sides of the transversal? Wait, no. Wait, angle 2 and angle 8: Let's consider the angles. Angle 2 and angle 5 are corresponding (congruent), angle 5 and angle 8: angle 5 and angle 8 are supplementary? No, angle 5 and angle 6 are supplementary (linear pair), angle 6 and angle 7 are vertical, angle 7 and angle 8 are supplementary. Wait, maybe a better approach: angle 2 and angle 8. Let's see, angle 2 and angle 4 are vertical (congruent), angle 4 and angle 8: are they corresponding? Angle 4 and angle 8: angle 4 is on line \(g\), below \(f\), angle 8 is on line \(h\), below \(f\), and they are in corresponding positions (since \(g\parallel h\)), so angle 4 and angle 8 are corresponding (congruent). But angle 2 and angle 4 are vertical (congruent), so by transitivity, angle 2 and angle 8: wait, no. Wait, angle 2 and angle 5 are corresponding (congruent), angle 5 and angle 8: angle 5 and angle 8 are same - side interior? No, angle 5 is above \(f\) on line \(h\), angle 8 is below \(f\) on line \(h\), so they are supplementary. Wait, maybe I made a mistake. Wait, the lines \(g\) and \(h\) are parallel, transversal \(f\). Angle 2 and angle 8: let's look at the diagram. Angle 2 is above \(f\) on line \(g\), angle 8 is below \(f\) on line \(h\). Wait, angle 2 and angle 3 are supplementary (linear pair), angle 3 and angle 8: angle 3 is below \(f\) on line \(g\), angle 8 is below \(f\) on line \(h\), and they are alternate - interior angles (since \(g\parallel h\)), so angle 3 and angle 8 are congruent. Then angle 2 and angle 3 are supplementary, so angle 2 and angle 8: angle 2 + angle 3=180°, and angle 3 = angle 8, so angle 2 + angle 8 = 180°? No, that can't be. Wait, no, maybe the diagram is different. Wait, in the diagram, angle 2 is above \(f\) between \(g\) and \(f\), angle 8 is below \(f\) between \(h\) and \(f\). Wait, maybe angle 2 and angle 8: let's check the options. The options are greater than, less than, congruent to. Since \(g\parallel h\), angle 2 and angle 8: are they congruent? Wait, angle 2 and angle 5 are corresponding (congruent), angle 5 and angle 8: angle 5 is above \(f\) on \(h\), angle 8 is below \(f\) on \(h\), so angle 5 and angle 8 are supplementary. So angle 2 (congruent to angle 5) and angle 8: angle 2 + angle 8=180°? No, that's not right. Wait, maybe I misread the diagram. Wait, the diagram: line \(f\) is horizontal, line \(g\) is a line with angles 1,2,3,4 (1 and 2 above \(f\), 4 and 3 below \(f\)), line \(h\) is parallel to \(g\), with angles 5,6,7,8 (5 and 6 above \(f\), 8 and 7 below \(f\)). So angle 2 is above \(f\), between \(g\) and \(f\), angle 8 is below \(f\), between \(h\) and \(f\). Now, angle 2 and angle 3 are supplementary (linear pair), angle 3 and angle 8: angle 3 is below \(f\) on \(g\), angle 8 is below \(f\) on \(h\), and since \(g\parallel h\), angle 3 and angle 8 are alternate - interior angles, so they are congruent. So angle 3 = angle 8. Then angle 2 + angle 3=180° (linear pair), so angle 2 + angle 8=180°? No, that would mean they are supplementary, not congruent. Wait, but the first part says angle 4 and angle 8 are corresponding angles (which are congruent). Angle 4 is below \(f\) on \(g\), angle 8 is below \(f\) on \(h\), so they are corresponding, so congruent. Angle 2 and angle 4 are vertical angles (congruent), so angle 2 = angle 4, and angle 4 = angle 8 (corresponding), so angle 2 = angle 8. Oh! I see, I made a mistake in the position of angle 4. Angle 4 is below \(f\) on \(g\), same as angle 8 is below \(f\) on \(h\). So angle 4 and angle 8 are corresponding (congruent). Angle 2 and angle 4 are vertical angles (congruent), so by transitivity, angle 2 and angle 8 are congruent.

Step2: Conclusion

Since \(g\parallel h\), angle 2 and angle 8: angle 2 is congruent to angle 8 because angle 2 and angle 4 are vertical angles (congruent), and angle 4 and angle 8 are corresponding angles (congruent, as \(g\parallel h\) cut by transversal \(f\)).

Answer:

congruent to