Introduction to Euclid’s Geometry Class 9 Maths Worksheet with Answers Chapter 5

Introduction to Euclid’s Geometry is the next chapter of the NCERT book we are handling here. The worksheet for Chapter 5 is organised into 4 sections: Basic, Standard, Advance, and HOTS. Solve the question in a section-wise format (Basic → HOTS) to avoid “axiom shock”. Attempt only after you have finished the chapters. It is important to understand the definitions, axioms, and postulates before attempting the worksheet questions. Below are the major topics covered in the chapter –

• Geometry meaning and origin
• Point line plane
• Axioms and postulates
• Euclid’s five postulates
• Reasoning using axioms

Class 9 Maths Worksheet – Chapter 5: Introduction to Euclid’s Geometry

Basic

1. Write the meaning of the word geometry (in 1 line).
2. Fill in the chain (in order):
   Solid → ____ → ____ → ____
3. Write the dimensions of:
   • Solid
   • Surface
   • Line
   • Point
4. From Euclid’s geometry, name the three undefined terms used in modern geometry.
5. State Euclid’s Postulate 1 in your own words.
6. True/False (write one-line reason):
   • Only one line can pass through a single point.
   • A terminated line can be produced indefinitely.
   • All right angles are equal.

Standard

1. Write Euclid’s common notions (axioms) for:
   • If equals are added to equals…
   • The whole is greater than…
2. Convert to modern statement:
   “Given two distinct points, there is a unique line that passes through them.”
   Write it as Axiom 5.1 type statement in your notebook.
3. Explain in 2–3 lines:
   Why do mathematicians avoid defining every term in geometry?
4. In a line segment AC, point B lies between A and C.
   Using Euclid’s axiom “things which coincide are equal”, write the result for:
   AB + BC = ____
5. Write the steps (only steps, no proof) to construct an equilateral triangle on a given line segment AB using circles.
6. Decide whether each is an axiom or a postulate:
   • A circle can be drawn with any centre and radius.
   • Things equal to the same thing are equal to one another.
   • A terminated line can be produced indefinitely.

Advance

1. Using Euclid’s axioms, justify (in 2–3 lines):
   If AB = PQ and PQ = XY, then AB = XY.
2. Two lines l and m intersect at two distinct points P and Q.
   Explain why this is impossible using the “unique line through two points” idea.
3. If a point C lies between A and B such that AC = BC, prove that:
   AC = 1/2 AB
   (Write a short proof.)
4. In a straight line A–B–C–D (points in this order), if AC = BD, prove that:
   AB = CD
5. Consider these two “postulates”:
   • Given two distinct points A and B, there exists a point C between A and B.
   • There exist at least three points not on the same line.

   Write:

   • Any undefined terms used
   • Whether they look consistent (Yes/No with reason)
6. Define (in your own words):
   • Parallel lines
   • Perpendicular lines
   • Line segment

   Also mention one term that you need to “already know” to define each.

HOTS

1. Euclid defined “a point is that which has no part”.
   Explain the problem with such definitions in 3–4 lines (hint: chain of definitions).
2. Postulate 5 talks about interior angles being less than two right angles.
   If the sum of interior angles on one side is less than 180°, what must happen to the two lines?
   (Write the statement in 1–2 lines.)
3. Give a real-life example to show:
   “Geometry started from the need to measure.”
   (Write one example in 2 lines.)
4. A student says: “Axioms are proved statements and theorems are assumed truths.”
   Correct the student in 2–3 lines.
5. Why is “The whole is greater than the part” considered a universal truth?
   Give one simple numeric example.
6. Create your own “mini-proof” using Euclid’s axiom:
   If equals are subtracted from equals, the remainders are equal.
   Take any simple numbers and write a 2-line proof-like statement.

Answer Key

Basic – Answers

1. Ans: Geo = earth, metrein = measure; geometry means measuring the earth/land.
   Hint: Write it in one simple sentence.
2. Ans: Solid → Surface → Line → Point
   Hint: Each step loses one dimension.
3. Ans:
   • Solid: 3D
   • Surface: 2D
   • Line: 1D
   • Point: 0D
4. Ans: Point, line, plane
   Hint: These are accepted without formal definition.
5. Ans: A straight line can be drawn joining any two points.
   Hint: “From one point to any other point.”
6. Ans:
   • False: infinitely many lines can pass through one point.
   • True: a segment can be extended on both sides.
   • True: all right angles are equal.

   Hint: Through one point, rotate the line in many directions.

Standard – Answers

1. Ans:
   • If equals are added to equals, the wholes are equal.
   • The whole is greater than the part.
2. Ans: Given two distinct points, exactly one line passes through both points.
   Hint: Emphasise “unique”.
3. Ans: If we define every word, each definition needs new words to be defined.
   This creates an endless chain, so some basic terms are kept undefined.
   Hint: Mention “chain of definitions”.
4. Ans: AB + BC = AC
   Hint: AB and BC together make AC.
5. Ans (steps):
   • Draw segment AB.
   • With centre A and radius AB, draw a circle.
   • With centre B and radius BA, draw another circle.
   • Let circles intersect at C.
   • Join AC and BC to get ΔABC.
6. Ans:
   • Circle with any centre and radius: Postulate
   • Equal to same thing: Axiom
   • Terminated line produced indefinitely: Postulate

Advance – Answers

1. Ans: If AB = PQ and PQ = XY, then AB = XY.
   Hint: Use axiom: “Things equal to the same thing are equal to one another.”
2. Ans: If two lines pass through both P and Q, then there are two different lines through the same two points.
   This contradicts the unique line through two points, so it’s impossible.
   Hint: This is the base idea of Theorem 5.1.
3. Ans: Since C lies between A and B, AB = AC + CB.
   Given AC = BC, so AB = AC + AC = 2AC.
   Hence AC = 1/2 AB.
   Hint: Replace BC with AC.
4. Ans: In order A–B–C–D:
   AC = AB + BC and BD = BC + CD.
   Given AC = BD ⇒ AB + BC = BC + CD ⇒ AB = CD.
   Hint: Subtract BC from both sides.
5. Ans:
   • Undefined terms: point, between, line (not formally defined here).
   • Consistent: Yes (they do not contradict each other).

   Hint: Consistent means “no contradiction”.

6. Ans (sample):
   • Parallel lines: lines in a plane that do not meet (needs idea of plane/line).
   • Perpendicular lines: lines meeting at 90° (needs idea of right angle).
   • Line segment: part of a line with two endpoints (needs idea of point/line).

HOTS – Answers

1. Ans: “No part” needs the meaning of “part”.
   If we define “part”, we may need new definitions again.
   This can go on forever, so some terms are left undefined.
   Hint: Mention “endless chain”.
2. Ans: The two lines, if extended, must meet on that side where the angle sum is less than 180°.
   Hint: This is exactly Postulate 5.
3. Ans (example): Measuring land plots after a flood erased boundaries, or planning roads/buildings using lengths and angles.
   Hint: One real situation is enough.
4. Ans: Axioms/postulates are assumed without proof.
   Theorems are proved using axioms, definitions, and logic.
   Hint: Swap the student’s statement.
5. Ans: It works for any quantities, so it is universal.
   Example: 9 is greater than 5 because 9 = 5 + 4.
   Hint: Show whole = part + something.
6. Ans (sample):
   If 15 = 15 and 7 = 7, then 15 − 7 = 15 − 7, so 8 = 8.
   Hint: Use same subtraction on equal numbers.

Worksheet for Other Class 9 Maths Chapters

• Number Systems Class 9 Maths Worksheet Chapter 1
• Polynomials Class 9 Maths Worksheet Chapter 2
• Linear Equations in Two Variables Class 9 Maths Worksheet Chapter 4
• Heron’s Formula Class 9 Maths Worksheet Chapter 10
• Surface Areas and Volumes Class 9 Maths Worksheet Chapter 11
• Statistics Class 9 Maths Worksheet Chapter 12