1. Problem 41: What is the measure of angle A in \(\triangle ABC\)? - Statement (1): \(\angle B = 60^\circ\) - Statement (2): \(\angle C = 30^\circ\) Step 1: Recall the triangle angle sum property: \(\angle A + \angle B + \angle C = 180^\circ\). Step 2: Using statement (1) alone, \(\angle B = 60^\circ\), but \(\angle C\) is unknown, so \(\angle A\) cannot be determined uniquely. Step 3: Using statement (2) alone, \(\angle C = 30^\circ\), but \(\angle B\) is unknown, so \(\angle A\) cannot be determined uniquely. Step 4: Using both statements together, \(\angle A = 180^\circ - 60^\circ - 30^\circ = 90^\circ\). Conclusion: Both statements together are sufficient, but neither alone is sufficient. Answer: (c). 2. Problem 42: Is triangle ABC equilateral? - Statement (1): \(AB = BC\) - Statement (2): \(\angle A = 60^\circ\) Step 1: Statement (1) alone tells us two sides are equal, so triangle is isosceles, but not necessarily equilateral. Step 2: Statement (2) alone tells us one angle is 60°, but other angles and sides are unknown, so cannot confirm equilateral. Step 3: Combining both statements, knowing one angle is 60° and two sides equal does not guarantee all sides equal or all angles 60°. Conclusion: Both statements together are not sufficient. Answer: (e). 3. Problem 43: What is the length of the radius of circle O? - Statement (1): Diameter is 10 cm. - Statement (2): Circumference is 20 cm. Step 1: Using statement (1), radius \(r = \frac{diameter}{2} = \frac{10}{2} = 5\) cm. Step 2: Using statement (2), circumference \(C = 2\pi r = 20\) cm, so \(r = \frac{20}{2\pi} = \frac{10}{\pi} \approx 3.18\) cm. Step 3: The two statements give different radius values, so each alone is sufficient to find radius, but they contradict each other. Conclusion: Each statement alone is sufficient. Answer: (d). 4. Problem 44: Are lines L1 and L2 parallel? - Statement (1): Alternate interior angles are congruent. - Statement (2): Corresponding angles are supplementary. Step 1: Statement (1) says alternate interior angles are equal, which is a property of parallel lines, so lines are parallel. Step 2: Statement (2) says corresponding angles are supplementary, but for parallel lines corresponding angles are equal, not supplementary. Step 3: Statement (2) alone is not sufficient to conclude lines are parallel. Conclusion: Statement (1) alone sufficient, statement (2) alone not sufficient. Answer: (a).