When a problem provides a relationship between two variables (like \(x+y=120\)) but doesn't give any other constraints, you usually cannot determine the relationship between the individual variables. Test a few valid pairs of numbers to confirm.
The relationship cannot be determined from the information given
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The Correct Option isD
Solution and Explanation
Step 1: Understanding the Concept:
This problem involves the properties of parallel lines and the angles formed by transversals. Specifically, it uses the concept of alternate interior angles and the sum of angles in a triangle. Step 2: Key Formula or Approach:
1. When two parallel lines are intersected by a transversal, alternate interior angles are equal.
2. The sum of the interior angles of a triangle is \(180^{\circ}\). Step 3: Detailed Explanation:
Let's analyze the figure. We have two parallel lines and two transversals that intersect to form a triangle between the parallel lines.
The angle marked \(x\) is an exterior angle. Its alternate interior angle is one of the angles inside the triangle. Let's call this angle \(\angle A\). So, \(\angle A = x\).
Similarly, the angle marked \(y\) is an exterior angle. Its alternate interior angle is another angle inside the triangle. Let's call this angle \(\angle B\). So, \(\angle B = y\).
The third angle in the triangle is given as \(60^{\circ}\).
The sum of the angles in this triangle must be \(180^{\circ}\).
\[
\angle A + \angle B + 60^{\circ} = 180^{\circ}
\]
Substituting \(x\) and \(y\):
\[
x + y + 60^{\circ} = 180^{\circ}
\]
Subtracting \(60^{\circ}\) from both sides gives us a relationship between \(x\) and \(y\):
\[
x + y = 120^{\circ}
\]
Step 4: Comparing the Quantities:
We need to compare \(x\) and \(y\). All we know is that their sum is 120. The individual values of \(x\) and \(y\) are not fixed. Scenario 1: If the triangle were isosceles with the two base angles being equal, then \(x = y\). In this case, \(x = y = 60\). (A = B) Scenario 2: It is possible that \(x = 40\) and \(y = 80\). Both are positive angles, and their sum is 120. In this case, \(y \textgreater x\). (B \textgreater A) Scenario 3: It is also possible that \(x = 90\) and \(y = 30\). Their sum is 120. In this case, \(x \textgreater y\). (A \textgreater B)
Since the relationship between \(x\) and \(y\) can change, it cannot be determined from the information given.
