Question 1
Question: Give a definition for a) parallel lines b) perpendicular lines Are these other terms that need to be defined first?
Answer: a) Given any straight line and a point not on it, there exists one and only one line which passes through that point and never intersects the given line, no matter how far they are extended.
b) Lines are said to be perpendicular to each other if they intersect a right angles. Since the definition of parallel lines is derived from Postulate 5, there are no terms which need to be defined first.
In the definition of perpendicular lines, Euclid has already define the terms like 'lines', 'right angles' and 'intersect'.
Question 2
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 infinite number of lines which pass through two distinct points.
iii) A terminated line can be produced indefinitely on both sides.
iv) If two circles are equal, then their radii are equal.
v) If AB = PQ and PQ = XY then AB = XY.
Answer: i) False. We know that a unique line can be drawn by joining two points, we can visualize that through a single point, infinite number of lines can be drawn.
ii) False. Given two distinct points, there is a unique line which passes through them.
iii) True. It is one of the postulates of Euclid (Postulate 2)
iv) True. If two circles are equal, the region bounded by one circle coincides with the region bounded by the other circle. Therefore, their radii also coincide.
v) True. According to one of Euclid's axioms, things which are equal to same thing are equal to each other.
Question 3
Question: Give the definition of the following terms. Are there other terms that need to be defined first
i) line segment
ii) radius of a circle
iii) square
Answer: Line segment: A line segment is a part of a line with two terminal points. The other terms that need to be defined in this definition. 'part', 'terminal point'
Radius of a circle: The distance from the centre of a circle to any point on the circle is its radius.
The other terms that need to be defined are 'centre', 'distance' and circle.
Square In a circle of any radius, draw to perpendicular diameters. The closed figure obtained by joining the points of intersection of the circle and the diameters is a square. The terms that need to be defined here are 'diameter', 'closed figure'
The assumption made are 'angle on a semicircle is a right angle' 'the sides of the square are equal'
Question 4
Question: If A, B and C are three points on a line, and B lies between A and C, then prove that AB + BC = AC.
Answer:
In the given figure AC coincides with AB + BC. According tone of Euclid's axioms, things which coincide with one another, are equal to one another.
According to axiom 5.1, a unique line passes through two given points. B lies on the line joining A and C.
Question 5
Question: Prove that an equilateral triangle can be constructed on any given line segment.
Answer:
Let AB be a given line segment. According to Euclid's postulate 1, a circle can be drawn with any centre and any radius.
Taking A as the centre and AB as radius, draw a circle.
Taking B as the centre and BA as radius, draw another circle. Let the two circles meet at a point, say C. Draw the line segments AC and BC.
Now AC = BC (Radius of the same circle) BC = AB (Radius of the same circle)
By Euclid's axiom things which are equal to the same thing are equal to one another. AC = BC = AB So, ABC is an equilateral triangle.
Question 6
Question: If a point C lies between two points A and B such that AC = BC, then prove that 
Answer:
AC = BC (given)
AC + AC = BC + AC (equals are added to equals, the wholes are equal)
(BC + AC = AB because things which coincide with one another are equal to one another)
(Things which are halves of same things are equal to one another)
Note that we have proved the above statements by using Euclid's axioms.
Question 7
Question: Prove that every line segment has one and only one mid point.
Answer: C is the mid point of AB.
Let us assume that there are two mid points of AB. Let the two mid points be C and D.
C is the mid point of AB
(Equals are added to equals, wholes are equal)
(Things which coincide one another, are equal to one another)
(Things which are halves of same thing are equal to one another)
Similar since D is a mid point of AB
From (1) and (2)
AC = AD (Things which are equal to same thing are equal to one another)
AC = AD and C and D are on the same line AB, implies C and D coincide. Every line has one and only one mid point.
Question 8
Question: In the given figure, if AC = BD, then prove that AB = CD.
Answer: AC = BD (given) ... (1)
AB + BC = AC (Point B lies between A and C) ... (2)
BC + CD = BD (Point C lies between B and D) ... (3)
Substituting (2) and (3) in (1), we have
AB + BC = CD + BC
(Substituting equals from equals are equal to one another)
Question 9
Question: Why is the Euclid's axiom given two distinct points, there is a unique line passes through them considered a universal truth.
Answer: Since it can be visualized for any thing in any part of the world, this is a universal truth.
Question 10
Question: Though Euclid did not require his fifth postulate to prove his first 28 theorem, why is this postulate most significant?
Answer: Euclid and many other mathematicians thought that the fifth postulate can be proved by using just the four postulates and the axioms. All the attempts to prove 5th postulate failed. But these efforts have led to the creation of non-Euclidean geometry. Till then everyone believed that Euclid's was the only geometry. Mathematicians are under the impression that world itself is Euclidean, but now the geometry of the universe we live in, has been shown to be a non-Euclidean geometry. It is also known as spherical geometry.
