Question

s, p and q are interior angles of an Isosceles triangle. Find the value of q.
1. s = 72°.
2. p and q are base angles of the triangle.

• 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 asked, but NEITHER statement ALONE is sufficient to answer the question ask.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hint:

When dealing with isosceles triangles, always consider the different cases for which angles are the equal base angles unless specified. Statement (2) is crucial here because it clarifies which angles are the base angles.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for the value of angle q in an isosceles triangle. An isosceles triangle has two equal sides and two equal base angles. The sum of interior angles in any triangle is 180°.
Step 2: Key Formula or Approach:
1. Sum of angles: \( s + p + q = 180° \)
2. Isosceles property: Two of the three angles must be equal.
Step 3: Detailed Explanation:
Analyze Statement (1): "s = 72°."
This gives us one angle. Since the triangle is isosceles, there are two possibilities for the other angles:
Case A: s is the vertex angle (the unique angle). The other two angles, p and q, are the equal base angles. \[ p = q = \frac{180° - 72°}{2} = \frac{108°}{2} = 54° \] In this case, q = 54°.
Case B: s is one of the base angles. Then another angle is also 72°. If p is the other base angle, then \( p = s = 72° \). The third angle would be \( q = 180° - 72° - 72° = 36° \).
If q is the other base angle, then \( q = s = 72° \). The third angle would be \( p = 180° - 72° - 72° = 36° \).
Since q could be 54° or 72° (or 36°, depending on which angle pairs are equal), we cannot find a unique value for q. Thus, Statement (1) is not sufficient.
Analyze Statement (2): "p and q are base angles of the triangle."
This tells us that p and q are the two equal angles in the isosceles triangle. So, \( p = q \). The sum of angles is \( s + p + q = 180° \), which becomes \( s + 2q = 180° \). Since we do not know the value of s, we cannot find a unique value for q. Thus, Statement (2) is not sufficient.
Analyze Statements (1) and (2) Together:
From (1), we know \( s = 72° \).
From (2), we know that p and q are the base angles, so \( p = q \). This means s must be the vertex angle. We can now substitute the value of s into the equation from our analysis of Statement (2):
\[ s + 2q = 180° \] \[ 72° + 2q = 180° \] \[ 2q = 180° - 72° \] \[ 2q = 108° \] \[ q = 54° \] This gives a single, unique value for q. Therefore, the statements together are sufficient.
Step 4: Final Answer:
Since neither statement alone is sufficient, but both together are, the correct option is (C).