Consider the following statements about circles and segments: (1) The number of circles that can be drawn through three non collinear points is infinity. (2) The angle formed in the minor segment of a circle is acute. Which of the above statements is or are correct?

Introduction / Context:
This conceptual question is about basic properties of circles: how many circles can pass through given points, and the nature of angles in segments of a circle. It tests whether you correctly recall some fundamental uniqueness and angle properties in circle geometry.

Given Data / Assumptions:

• Statement 1: The number of circles that can be drawn through three non collinear points is infinity.
• Statement 2: The angle formed in the minor segment of a circle is acute.
• Non collinear points means the three points do not lie on a straight line.
• A segment of a circle is the region bounded by a chord and the corresponding arc. The smaller region is the minor segment, and the larger is the major segment.

Concept / Approach:
We analyse each statement using well known results:

• Through any three non collinear points, there exists exactly one circle, called the circumcircle of the triangle formed by those points.
• Angles subtended by a chord at points on the same arc have a fixed relation. The inscribed angle subtended by a chord on the same side as the minor arc is obtuse, while on the side of the major arc it is acute. Thus, the angle in the minor segment is typically obtuse, not acute.

Step-by-Step Solution:
Step 1: Consider the three non collinear points as vertices of a triangle. Step 2: In geometry, it is a standard theorem that there is a unique circle passing through the three vertices of a triangle. This circle is called the circumcircle of the triangle. Step 3: Therefore, Statement 1, which says there are infinitely many circles through three non collinear points, is false. There is exactly one. Step 4: Now consider Statement 2 about angles in a minor segment. Step 5: Take a chord AB of a circle. The minor segment is the smaller region bounded by chord AB and the minor arc AB. Any angle with vertex on the minor arc and intercepting chord AB actually subtends the major arc. Step 6: Since the angle at the circumference equals half the measure of the intercepted arc, an angle intercepting a major arc (greater than 180 degrees) is obtuse. Step 7: Hence, angles formed in the minor segment of the circle are typically obtuse, not acute. Step 8: Therefore, Statement 2, which claims such an angle is acute, is also false. Step 9: Since both statements are false, the correct option is that neither 1 nor 2 is correct.

Verification / Alternative check:
You can verify Statement 1 by constructing different sets of three non collinear points and trying to draw more than one circle through them; all attempts will coincide with a single unique circle. For Statement 2, a careful diagram of a chord and its arcs will show that the angle in the major segment is acute, while the angle in the minor segment is obtuse, confirming the reasoning above.

Why Other Options Are Wrong:
1 only or 2 only: These would require one of the statements to be true, but we have seen that both contradict standard theorems about circles and segments.
Both 1 and 2: This is impossible since both are individually false.
Cannot be determined: The correctness of the statements follows from well established geometric facts and does not require any numerical data, so it can be determined.

Common Pitfalls:
Learners often confuse the case of two points (infinitely many circles through a fixed chord) with three non collinear points (only one circle). Another common error is mixing up the angle in the minor segment with the angle in the major segment. Keeping a clear mental or drawn diagram of chords, arcs and segments helps avoid these misunderstandings.

Final Answer:
Both statements are false, so the correct choice is Neither 1 nor 2.