RS Aggarwal Class 9 Mathematics Solutions Chapter 3 Introduction to Euclids Geometry

Question 1. Distinguish between a theorem and an axiom.
Answer: A theorem is a mathematical claim that requires proof to be established as true. An axiom, by contrast, is a fundamental truth accepted without proof. The Pythagoras Theorem serves as an example of a theorem, while the statement that a unique line can be drawn through any two points illustrates an axiom.
In simple words: A theorem must be proved, but an axiom is just accepted as true from the start.

Exam Tip: Always remember - theorems need proof, axioms do not. Use concrete examples (Pythagoras, parallel lines) to strengthen your answer.

Question 2. Define the following geometric terms: (i) Line segment (ii) Ray (iii) Intersecting Lines (iv) Parallel Lines (v) Half-line (vi) Concurrent lines (vii) Collinear points (viii) Plane.
Answer:
(i) Line segment: The straight path connecting two points is called a line segment.
(ii) Ray: When a line segment is extended indefinitely in one direction, it forms a ray.
(iii) Intersecting Lines: Two lines that meet at a common point are called intersecting lines, meaning they share a common point.
(iv) Parallel Lines: Two lines in a plane are considered parallel if they share no common point and never meet.
(v) Half-line: A ray that excludes its initial point is called a half-line.
(vi) Concurrent lines: Three or more lines that intersect at the same point are said to be concurrent.
(vii) Collinear points: Three or more points are collinear if they all lie on the same line.
(viii) Plane: A plane is a surface with the property that every point on the line joining any two points on it also lies on it.
In simple words: A line segment has two endpoints. A ray has one starting point and goes forever in one direction. Parallel lines never meet. Concurrent lines all meet at one spot. Collinear points sit on the same straight line.

Exam Tip: Learn all eight definitions thoroughly - they form the foundation of geometry. Use sketches to visualize each concept.

Question 3. From the given figure, identify: (i) Six points (ii) Five line segments (iii) Four rays (iv) Four lines (v) Four collinear points.
Answer:
(i) Six points: A, B, C, D, E, F
(ii) Five line segments: EG, FH, EF, GH, MN
(iii) Four rays: EP, GR, GB, HD
(iv) Four lines: AB, CD, FQ, RS
(v) Four collinear points: M, E, G, B
In simple words: Points are specific locations marked on the figure. Line segments join two points. Rays start at one point and extend forever. Collinear points all sit on the same straight line.

Exam Tip: When identifying geometric elements from a figure, mark each element carefully to avoid confusion between rays, line segments, and full lines.

Question 4. State any three postulates or axioms related to lines and points.
Answer:
(i) Infinitely many lines can pass through a single given point.
(ii) Exactly one unique line passes through any two given points.
(iii) Two distinct lines intersect at most at a single point, not more.
In simple words: You can draw unlimited lines through one point. But only one line goes through two different points. Two lines meet at just one spot.

Exam Tip: These are foundational axioms - memorize them exactly and recall them when answering questions about line existence and intersection properties.

Question 5. State whether the following statements are true or false: (i) A line has no length. (ii) A ray has two end points. (iii) A line segment has infinite length. (iv) Two distinct lines can meet at two points. (v) A ray has one end point. (vi) A line segment joining two points is the shortest distance between those points. (vii) Infinite number of lines can pass through a given point. (viii) Only one line can pass through two given points. (ix) Two intersecting lines cannot both be parallel to a third line. (x) A plane is a two-dimensional figure. (xi) A plane has a boundary. (xii) A line is one-dimensional.
Answer:
(i) False
(ii) False
(iii) False
(iv) False
(v) True
(vi) True
(vii) True
(viii) True
(ix) True
(x) False
(xi) False
(xii) True
In simple words: Remember - a line extends forever with no endpoints, a ray starts at one point and goes on forever, and a line segment has two definite endpoints and finite length. A plane is infinite with no boundary.

Exam Tip: Distinguish carefully between lines (infinite in both directions), rays (one endpoint), and line segments (two endpoints). This is tested frequently.