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Which of the lines in the figure are parallel? Which is the transversal?
Parallel lines never intersect. We know that. Transversals are lines that intersect two or more (often parallel) lines. We can identify a and b as the parallel lines (if not by their lack of intersection, then by the helpful arrowheads along the lines themselves) and c as the transversal (because it crosses the other two).
Which angles are congruent in this figure?
Just by looking at the image, we can see that ∠1 is definitely acute and angles ∠2 and ∠3 are definitely not. In fact, ∠2 and ∠3 are vertical angles, which means they're congruent. We can also see that ∠1 and ∠4 are congruent because the vertical angle to ∠4 corresponds to ∠1. Since both corresponding and vertical angles are congruent, we can conclude that ∠1 ≅ ∠4 because they're alternate interior angles.
What is the relationship between ∠1 and ∠3?
They're neither vertical nor corresponding angles. One is interior and the other is exterior, and they're not even on the same side of the transversal. Do these guys even have a name that applies to them?
In short, yes. It's called being supplementary. We can see that because ∠1 ≅ ∠4 (since they're alternate interior angles), and ∠4 is supplementary to ∠3 because together they add up to a straight angle. By substitution, ∠1 is supplementary to ∠3 as well.
If ∠3 has a measure of 138°, what is the measure of an angle adjacent to ∠1?
Adjacent angles share both a side and a vertex. But…∠1 has two adjacent angles: one that's still in between the parallel lines and on the other side of the transversal (∠5) and another that's on the same side of the transversal, but exterior to the two parallel lines (∠6). Which one are we looking at?
An even better question: does it matter? They're vertical angles, so they're congruent to each other anyway. We can compare both the angles to ∠3 just to be sure: ∠5 and ∠3 are corresponding angles, and ∠6 and ∠3 are alternate exterior angles. No matter what way we look at it, any angle adjacent to ∠1 will be congruent to ∠3—and therefore be 138° in measure.