Answer: Option B: \(y = 30\)
Step-by-step explanation: The problem in the image is asking which relationship between \(x\) and \(y\) is sufficient to prove that lines \(j\) and \(k\) are parallel.
Lines are parallel if their corresponding angles are equal or if their alternate interior angles are equal.
Given the angles on lines \(j\) and \(k\):
- On line \(j\), the angle is \(m < (x + y)\degree\).
- On line \(k\), one angle is \((x + 30)\degree\) and another is \((3x - y - 60)\degree\).
We can check each option to see if it makes the corresponding angles or alternate interior angles equal:
- **Option A: \(x = 60\)**. Substituting \(x = 60\) into the angle expressions, we get:
- On line \(j\), the angle becomes \(60 + y\).
- On line \(k\), one angle becomes \(90\degree\) and another becomes \(120 - y\).
- These angles are not equal, so \(j\) is not parallel to \(k\) for this option.
- **Option B: \(y = 30\)**. Substituting \(y = 30\) into the angle expressions, we get:
- On line \(j\), the angle becomes \(x + 30\).
- On line \(k\), one angle becomes \(x + 30\degree\) and another becomes \(3x - 30\).
- The corresponding angles are equal, so \(j\) is parallel to \(k\) for this option.
- **Option C: \(y = x - 30\)**. Substituting \(y = x - 30\) into the angle expressions, we get:
- On line \(j\), the angle becomes \(2x - 30\).
- On line \(k\), one angle becomes \(x + 30\degree\) and another becomes \(2x - 30\).
- These angles are not equal, so \(j\) is not parallel to \(k\) for this option.
- **Option D: \(y = 150 - 2x\)**. Substituting \(y = 150 - 2x\) into the angle expressions, we get:
- On line \(j\), the angle becomes \(150 - x\).
- On line \(k\), one angle becomes \(x + 30\degree\) and another becomes \(150 - x\).
- These angles are not equal, so \(j\) is not parallel to \(k\) for this option.
Therefore, the correct answer is **Option B: \(y = 30\)**. This is the only option that makes the corresponding angles equal, which is a condition for lines to be parallel.
