Class 9 Mathematics
Chapter 5: Introduction to Euclid’s Geometry
Exercise 5.1 – Stepwise Solutions
Q1. Which of the following statements are true and which are false? Give reasons.
(i) Only one line can pass through a single point.
Answer: False
Infinitely many lines can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
Answer: False
Only one unique line can pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
Answer: True
A line segment can be extended endlessly in both directions.
(iv) If two circles are equal, then their radii are equal.
Answer: True
Equal circles have equal radii.
(v) If AB = PQ and PQ = XY, then AB = XY.
Answer: True
This follows from the transitive property of equality.
Q2. Definitions
(i) Parallel lines: Lines in the same plane that never intersect.
(ii) Perpendicular lines: Lines that intersect at 90°.
(iii) Line segment: A part of a line with two endpoints.
(iv) Radius of a circle: The distance from the centre to any point on the circle.
(v) Square: A quadrilateral with all sides equal and all angles 90°.
Undefined terms needed: Point, Line, Plane
Q3. Postulates
(i) There exists a point between two distinct points.
(ii) There exist at least three non-collinear points.
Answer:
Yes, they contain undefined terms like point, line, between.
Yes, they are consistent.
Yes, they follow Euclid’s postulates.
Q4. Prove that AC = 1/2 AB
Given: C lies between A and B, and AC = BC
AB = AC + BC
AB = AC + AC
AB = 2AC
AC = 1/2 AB
Q5. Prove that every line segment has one and only one midpoint
A midpoint divides a line segment into two equal parts.
If there were two midpoints, it would contradict equality.
Hence, a line segment has exactly one midpoint.
Q6. Prove that AB = CD
Given: AC = BD
AC = AB + BC
BD = BC + CD
So, AB + BC = BC + CD
Subtract BC from both sides:
AB = CD
Q7. Why is Axiom 5 a universal truth?
Axiom 5 states that “The whole is greater than the part.”
This is true for all objects, not just geometrical figures.
Hence, it is called a universal truth.