Question

Which of the following statements are true and which are false? Give reasons for your answers. 

(i) Only one line can pass through a single point. 

(ii) There are an infinite number of lines which pass through two distinct points. 

(iii) A terminated line can be produced indefinitely on both the sides. 

(iv) If two circles are equal, then their radii are equal. 

(v) In Fig., if AB = PQ and PQ = XY, then AB = XY

Answer 1

(i) Only one line can pass through a single point.

False.
According to Euclid's first postulate, "A straight line can be drawn from any one point to any other point." This means that an infinite number of lines can pass through any given point. For example, you can draw many different lines that all pass through a single point, each with a different slope or direction.

(ii) There are an infinite number of lines which pass through two distinct points.

False.
According to Euclid's second postulate, "Two distinct points determine a unique straight line." This means that only one line can pass through two distinct points. If two points are given, there is exactly one line that can be drawn through them, not an infinite number of lines.

(iii) A terminated line can be produced indefinitely on both the sides.

True.
A terminated line (or line segment) is a portion of a line with two distinct endpoints. However, if you extend or "produce" the line segment indefinitely in both directions, it forms a line. According to the definition of a line, it has no endpoints and extends infinitely in both directions.

(iv) If two circles are equal, then their radii are equal.

True.
By definition, two circles are equal if they have the same radius and the same center. So, if two circles are equal, their radii must indeed be equal. The radius is the defining characteristic of a circle.

(v) In Fig., if AB = PQ and PQ = XY, then AB = XY.

True.
This statement follows the Transitive Property of Equality, which states that if AB = PQ and PQ = XY, then it must follow that AB = XY. This property holds true in mathematics and geometry, where if two quantities are both equal to a third quantity, they must be equal to each other.

Summary of Answers:

(i) False: An infinite number of lines can pass through a single point.

(ii) False: Only one line passes through two distinct points.

(iii) True: A terminated line can be produced indefinitely on both sides.

(iv) True: If two circles are equal, their radii are equal.

(v) True: If AB = PQ and PQ = XY, then AB = XY.