Question

In the figure above PRS is a triangle, what is the measure of the angle PSQ?

I. QS=QR=1
II. PR=2

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II together are sufficient to answer the question but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

In geometry problems, look for isosceles and equilateral triangles, as they provide powerful information about side lengths and angles. Break down complex figures into simpler triangles and apply the sum of angles theorem.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency problem in geometry. We need to find the measure of \(\angle PSQ\). We are given a larger triangle \(\triangle PRS\) with a point Q on the base PR. We also see \(\angle PQS = 120^\circ\) marked in the diagram.
Step 2: Detailed Explanation:
From the figure, \(\angle PQS\) and \(\angle RQS\) are on a straight line, so they are supplementary. \[ \angle RQS = 180^\circ - \angle PQS = 180^\circ - 120^\circ = 60^\circ \] Analyze Statement I: "QS=QR=1"
This tells us that \(\triangle QRS\) is an isosceles triangle with \(QS=QR\). In an isosceles triangle, the angles opposite the equal sides are equal. So, \(\angle QSR = \angle QRS\). The sum of angles in \(\triangle QRS\) is 180\(^\circ\). \[ \angle RQS + \angle QSR + \angle QRS = 180^\circ \] \[ 60^\circ + \angle QSR + \angle QSR = 180^\circ \] \[ 2 \times \angle QSR = 120^\circ \implies \angle QSR = 60^\circ \] This means \(\triangle QRS\) is an equilateral triangle. We now know all angles and sides of \(\triangle QRS\), but we have no information about \(\triangle PQS\) other than \(\angle PQS = 120^\circ\) and side \(QS=1\). We cannot determine \(\angle PSQ\). Statement I is not sufficient.
Analyze Statement II: "PR=2"
This gives the length of the base of the large triangle. This information alone does not help us determine any angles. Statement II is not sufficient.
Analyze Statements I and II Together:
From Statement I, we know \(QR = 1\). From Statement II, we know \(PR = 2\). Since Q lies on PR, we have \(PR = PQ + QR\). \[ 2 = PQ + 1 \implies PQ = 1 \] Now, consider \(\triangle PQS\). We know two sides, \(PQ = 1\) and \(QS = 1\), and the angle between them, \(\angle PQS = 120^\circ\). Since \(PQ = QS = 1\), \(\triangle PQS\) is an isosceles triangle. The angles opposite the equal sides must be equal. \[ \angle PSQ = \angle SPQ \] The sum of angles in \(\triangle PQS\) is 180\(^\circ\). \[ \angle PQS + \angle PSQ + \angle SPQ = 180^\circ \] \[ 120^\circ + \angle PSQ + \angle PSQ = 180^\circ \] \[ 2 \times \angle PSQ = 60^\circ \] \[ \angle PSQ = 30^\circ \] We have found a unique value for \(\angle PSQ\). Therefore, both statements together are sufficient.
Step 3: Final Answer:
Neither statement is sufficient on its own, but together they are sufficient. This corresponds to option (C).