8. Angles, Lines and Transversals
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Use the relationships between angles formed when a transversal cuts two lines.
See solution.
We are asked to explain how we can use angle measures to tell whether two lines are parallel. To do so, we use the relationship between the angles created when a transversal cuts the two lines. Let's consider two example lines a and b.
Let's cut the two lines with a transversal. This creates 8 angles.
Let's use each of these rules to confirm whether our two lines are parallel. First, we will check whether the same-side interior angles are supplementary. In our case, the pairs of same-side interior angles are ∠ 4 and ∠ 5, and ∠ 3 and ∠ 6.
Let's note the measures of these angles.
As we can see, in both cases the sum of the measures of the angles is 180^(∘). 83^(∘) + 97^(∘) = 180^(∘) This means that same-side interior angles are supplementary, so the lines are parallel. Now let's confirm that the lines are parallel using alternate interior angles. The pairs of alternate interior angles are ∠ 4 and ∠ 6, and ∠ 3 and ∠ 5.
Once again, we note the measures of these angles.
As we can see, both pairs of alternate interior angles are congruent. This confirms that the lines are parallel. Finally, we will use the angle measures of corresponding angles. Here, the pairs of corresponding angles are ∠ 1 and ∠ 5, ∠ 2 and ∠ 6, ∠ 3 and ∠ 7, and ∠ 4 and ∠ 8.
Note the measures of these angles.
All corresponding angles are congruent, which means that the lines are parallel.