Question 1: Which of the following statements are true and which are false? Give reasons for your answers.

1. Statement: Only one line can pass through a single point.
   Answer: False.
   Reason: Through a single point, an infinite number of lines can pass in different directions.
2. Statement: There are an infinite number of lines which pass through two distinct points.
   Answer: False.
   Reason: According to Euclid's Postulate 1, there is a unique line that passes through two distinct points.
3. Statement: A terminated line can be produced indefinitely on both sides.
   Answer: True.
   Reason: This is directly based on Euclid's Postulate 2, which states a terminated line (line segment) can be produced indefinitely to form a line.
4. Statement: If two circles are equal, then their radii are equal.
   Answer: True.
   Reason: If we superimpose one circle over another of equal area, their boundaries and centers will coincide. Thus, their radii will also coincide and be equal.
5. Statement: In the given figure, if AB = PQ and PQ = XY, then AB = XY.
   Answer: True.
   Reason: According to Euclid's Axiom 1, things which are equal to the same thing are equal to one another. Since both AB and XY are equal to PQ, they are equal to each other.

Question 2: Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?

Yes, there are basic terms like point, line, plane, ray, and angle that need to be understood as foundational terms before defining these.

• Parallel Lines: Two straight lines in the same plane that do not intersect each other, no matter how far they are extended. (Undefined terms needed: plane, intersect).
• Perpendicular Lines: Two lines that intersect each other at an angle of 90° (right angle). (Undefined terms needed: line, intersecting point, angle).
• Line Segment: A part of a line that has two fixed endpoints. (Undefined terms needed: line, point).
• Radius of a Circle: The distance from the center of a circle to any point on its boundary. (Undefined terms needed: circle, center, point).
• Square: A quadrilateral in which all four sides are equal and all four angles are right angles (90°). (Undefined terms needed: quadrilateral, side, angle).

Question 3: Consider two postulates given below:

(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line.

Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.

Solution:

• Undefined Terms: Yes, these postulates contain undefined terms such as 'point' and 'line'.
• Consistency: Yes, these postulates are consistent because they deal with two different physical situations:
  • Postulate (i) describes a situation where point C lies on the line segment joining points A and B, situated between them.
  • Postulate (ii) describes a situation where point C does not lie on the line passing through points A and B (i.e., they form a triangle).
• Relation to Euclid's Postulates: No, these postulates do not directly follow from Euclid's postulates. However, they follow from the axiom that "Given two distinct points, there is a unique line that passes through them."

Question 4: If a point C lies between two points A and B such that AC = BC, then prove that AC = ½ AB. Explain by drawing the figure.

Proof:

We are given that:

AC = BC —— (Equation 1)

Add AC to both sides of Equation 1 (According to Euclid's Axiom 2: If equals are added to equals, the wholes are equal):

AC + AC = BC + AC
2AC = BC + AC —— (Equation 2)

From the figure, we can observe that the point C lies between A and B. Thus, the segment AC and BC together coincide with the line segment AB:

BC + AC = AB

Substituting this in Equation 2, we get:

2AC = AB
AC = ½ AB

Hence proved.

Question 5: In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

Proof:

Let us assume by contradiction that the line segment AB has two midpoints, say C and D.

If C is the midpoint of AB, then:

AC = ½ AB —— (Equation 1)

If D is also a midpoint of AB, then:

AD = ½ AB —— (Equation 2)

Comparing Equation 1 and Equation 2:

AC = AD

According to Euclid's geometry, this is only possible if points C and D coincide with each other. Therefore, every line segment has one and only one midpoint.

Hence proved.

Question 6: In the given figure, if AC = BD, then prove that AB = CD.

Proof:

From the figure, we can write:

AC = AB + BC —— (Equation 1)
BD = BC + CD —— (Equation 2)

We are given that:

AC = BD

Substitute the values from Equation 1 and Equation 2:

AB + BC = BC + CD

Subtract BC from both sides (According to Euclid's Axiom 3: If equals are subtracted from equals, the remainders are equal):

AB + BC - BC = BC + CD - BC
AB = CD

Hence proved.

Question 7: Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)

Solution:

Euclid’s Axiom 5 states that "The whole is greater than the part."

This is considered a universal truth because it applies to any field of study, not just mathematics and geometry. For example:

• In geography, a country is a part of a continent, and the continent (the whole) is always larger than the country (the part).
• In physical chemistry, a molecule is a part of a compound, and the mass of the compound is always greater than the mass of its constituent molecules.
• In arithmetic, if you take any positive number (say 10) and take a part of it (say 3), the whole (10) will always be greater than the part (3).

Since this statement holds true for everything in the universe, it is considered a universal truth.