Are any of the lines in the picture parallel? How does this affect the relationship of the measures of ∠ 5 and ∠ 6?
No, see solution.
Practice makes perfect
When a transversal intersects two lines, four angles are created around each point of intersection. These four angles are called corresponding angles. Corresponding angles are angles that lie in the same position around the different intersection points.
The following pairs of angles are corresponding angles in this figure.
∠ 1 &and ∠ 5
∠ 2 &and ∠ 6
∠ 3 &and ∠ 7
∠ 4 &and ∠ 8
Keep in mind that the corresponding angles are congruentonly when a transversal intersects parallel lines. Now, we want to decide if our friend described the relationship between the two angles in the given diagram correctly. We will label the lines to help us refer to them later. Let's take a look at the given picture.
Now let's see what our friend suggests.
∠ 5 is congruent to ∠ 6.
Notice that ∠ 5 and ∠ 6 are corresponding angles. We know that corresponding angles are congruent when parallel lines are cut by a transversal. However, lines a and b do not appear to be parallel. Because of this, we cannot say that ∠ 5 is congruent to ∠ 6, so our friend is incorrect.
Extra
When Would Our Friend Be Correct?
We know that our friend is incorrect. Let's take a look at the graph for which their statement would be correct.
Notice that here lines a and b are parallel. This means that ∠ 5 and ∠ 6 are congruent.
