What is the measure of angle \( x \)?
Statement (I): \( PQ \parallel RS \).
Statement (II): Line \( AB \) bisects \( PQ \).
Show Hint
Whenever a figure contains parallel lines and a transversal, first look for corresponding angles, alternate interior angles, and supplementary angle pairs before attempting lengthy calculations.
Concept:
In Data Sufficiency questions, our objective is not necessarily to calculate the exact value of the unknown quantity but to determine whether the given statement(s) provide enough information to uniquely determine it.
For geometric figures involving parallel lines, angle relationships such as corresponding angles, alternate interior angles, co-interior angles, and vertically opposite angles can often be used to determine unknown angles.
Step 1: Analyze Statement (I) alone.
Statement (I) tells us that
\[
PQ \parallel RS
\]
When two parallel lines are intersected by a transversal, several angle relationships become fixed.
Using the angle information provided in the figure (such as the given \(60^\circ\) angle), the value of angle \(x\) can be uniquely determined through properties of parallel lines.
Since a unique value of \(x\) can be obtained, Statement (I) alone is sufficient.
Step 2: Analyze Statement (II) alone.
Statement (II) states that line \(AB\) bisects \(PQ\).
This only gives information about lengths or division of a line segment.
It does not provide any information regarding angle relationships, slopes of lines, or parallelism.
Therefore, many different geometric configurations are possible, producing different values of \(x\).
Hence Statement (II) alone is insufficient.
Step 3: Final conclusion.
Since Statement (I) alone determines the value of \(x\),
\[
\boxed{\text{Statement (I) alone is sufficient}}
\]
