Introduction to Euclid Geometry
EX 5.1 QUESTION 1.
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 figure,5.9 if AB – PQ and PQ = XY, then AB = XY.
(i) False, because there is an infinite number of lines that can pass through a single point.
(ii) False, because only one line that can be drawn through two distinct points.
(iii) True, because a terminated line can be produced indefinitely on both the sides
(iv) True, because if two circles are equal, then their radii are also equal.
(v) True, because according to Euclid’s axiom- “Things which are equal to the same thing are also equal to one another”.
EX 5.1 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?
(i) Parallel lines
(ii) Perpendicular lines
(iii) Line segment
(iv) Radius of a circle
(i) Parallel Lines: Two lines having the same common points. Parallel lines never intersect each other.
(ii) Perpendicular Lines: Lines which intersect each other in a plane at right angles are perpendicular lines.
(iii) Line Segment: The line which has two ends points.
(iv) The radius of a circle : The line segment from the centre of a circle is called the radius of the circle.
(v) Square : A quadrilateral having all sides equal and all angles are right angles is called a square.
EX 5.1 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.
Yes, these postulates contain undefined terms. Undefined terms in the postulates are:
(i) Given two points A and B, there is a point C lying on the line in between them.
(ii) Given points A and B, you can take point C not lying on the line through A and B.
These postulates do not follow from Euclid’s postulates, however, they follow from the axiom, “Given two distinct points, there is a unique line that passes through them.”
EX 5.1 QUESTION 4.
If a point C lies between two points A and B such that AC = BC, then prove that AC = AB, explain by drawing the figure.
Given AC = BC
∴ AC + AC = BC + AC [If equals added to equals]
or 2AC = AB [∵ AC + BC = AB]
or AC = 1/2 AB
EX 5.1 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.
AC = 1/2AB
and AD = 1/2AB
AC = AD [Things which are equal to the same thing are equal to one another.]
EX 5.1 QUESTION 6.
In figure, if AC = BD, then prove that AB = CD.
Given – AC = BD
⇒ AC- BC= BD – BC [When equals are subtracted from equals, remainders are equal]
⇒ AB = CD
EX 5.1 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.)
This is true in all the situations. This is a universal truth.