Question

In the figure above, PRS is a triangle, what is the measure of the angle PSQ? 
(1) \( QS = QR = 1 \) 
(2) \( PR = 2 \)

• A. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
• B. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
• C. Both statements 1 and 2 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 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

When working with geometry problems involving triangles, ensure that you have enough information about sides, angles, and relationships to solve the problem. Sometimes, more data is needed.

Answer 1

Answer: (E)

Step 1: Analyze Statement 1.
Statement 1 tells us that \( QS = QR = 1 \). This information gives us that triangle \( PQR \) is an isosceles triangle, but it does not provide enough information to calculate the measure of angle \( PSQ \) because we still don’t know the other angles or lengths in the figure. Hence, statement 1 alone is insufficient.
Step 2: Analyze Statement 2.
Statement 2 tells us that \( PR = 2 \). However, this information alone does not provide enough to calculate angle \( PSQ \) because it does not tell us anything about the other sides or angles in the triangle. So, statement 2 alone is also insufficient.
Step 3: Combine Both Statements.
Combining both statements does not give us enough information to answer the question because we are still missing key data, such as the angles in the triangle. Thus, additional data is needed.
Step 4: Conclusion.
The correct answer is (E). Statements 1 and 2 together are not sufficient to answer the question.