Welcome to AssamSchool. This article contains 30 Extra Questions from Introduction to Euclid’s Geometry from Class 9 Mathematics Chapter 5. These Introduction to Euclid’s Geometry Class 9 Extra Questions are helpful to test your understanding and knowledge of the chapter. These additional questions will not only help you to excel in the chapter but also will give you a strong foundation in Mathematics.
We have also provided step-by-step answers to the questions for reference. So, let’s start…
30 Introduction to Euclid’s Geometry Class 9 Extra Questions
1. State Euclid’s five postulates.
2. Define a “line segment” in Euclid’s terms.
3. True or False: A terminated line can be produced indefinitely. Justify.
4. What is the difference between an axiom and a postulate?
5. Name one contribution of Egyptian geometry mentioned in the chapter.
6. What are the undefined terms in Euclidean geometry?
7. True or False: All right angles are equal. Justify using Euclid’s postulates.
8. What does Euclid’s first postulate state?
9. How did the Indus Valley Civilisation use geometry?
10. What is the purpose of the Sulbasutras in ancient India?
11. Explain Euclid’s Axiom 4: “Things which coincide are equal.” Give an example.
12. Why are some terms like “point” and “line” left undefined in geometry?
13. Prove: Two distinct lines cannot intersect at more than one point.
14. Using Postulate 3, describe how to construct a circle with radius \( AB \).
15. What does Euclid’s Axiom 5 (“The whole is greater than the part”) mean? Provide an example.
16. How did Greeks differ from Egyptians in their approach to geometry?
17. If \( AC = BD \), prove \( AB = CD \) (refer to Fig. 5.10 in the textbook).
18. Explain why Euclid’s fifth postulate is more complex than the others.
19. True or False: A surface has only one dimension. Correct the statement if false.
20. How would you use Euclid’s axioms to show \( 2x = 2y \) implies \( x = y \)?
21. Prove that every line segment has exactly one midpoint using Euclid’s axioms.
22. Discuss the historical significance of Euclid’s Elements in systematizing geometry.
23. Why is the fifth postulate controversial? How does it differ from the first four?
24. Explain the statement: “A system of axioms must be consistent.”
25. Using Euclid’s postulates, prove that an equilateral triangle can be constructed on any line segment.
26. How does Euclid’s Axiom 1 (“Things equal to the same thing are equal”) apply to geometric figures?
27. If two circles intersect at two points, prove that their centers lie on the perpendicular bisector of the common chord.
28. Why did Euclid avoid defining terms like “length” and “breadth”?
29. Discuss the role of deductive reasoning in Euclid’s geometry.
30. Prove: If a point \( C \) lies between \( A \) and \( B \), then \( AC + CB = AB \).
Answers
- Answer:
- Postulate 1: A straight line can join any two points.
- Postulate 2: A terminated line can be extended indefinitely.
- Postulate 3: A circle can be drawn with any center and radius.
- Postulate 4: All right angles are equal.
- Postulate 5: If a line crosses two others and the interior angles sum to less than \(180°\), the lines meet on that side.
- Answer:
A line segment is a terminated line with two distinct endpoints. - Answer:
True. Postulate 2 states a terminated line (line segment) can be extended indefinitely. - Answer:
Axioms are universal truths in mathematics, while postulates are specific to geometry. - Answer:
Egyptians used geometry to rebuild field boundaries after the Nile floods and construct pyramids. - Answer:
Point, line, and plane. - Answer:
True. Postulate 4 states all right angles are equal. - Answer:
A straight line can be drawn between any two points. - Answer:
They built planned cities with parallel roads and standardized brick ratios \((4:2:1)\). - Answer:
They provided rules for constructing altars and fireplaces for Vedic rituals. - Answer:
If two geometric figures (e.g., triangles) coincide perfectly when placed on each other, they are equal. Example: Two circles with the same radius are congruent. - Answer:
To avoid infinite regression in definitions. These terms are intuitively understood. - Answer:
Assume two lines intersect at two points \( P \) and \( Q \). By Axiom 5.1, only one line passes through \( P \) and \( Q \). Contradiction. Hence, two lines cannot intersect at more than one point. - Answer:
Place the compass at point \( A \), extend it to \( B \), and draw the circle. - Answer:
A whole quantity (e.g., a line segment \( AB\)) is greater than any of its parts (e.g., \( AC \) where \( C \) is between \( A \) and \( B \)). - Answer:
Greeks focused on proofs and reasoning; Egyptians focused on practical applications. - Answer:
Given \( AC = BD \), subtract \( BC \) from both: \( AC – BC = BD – BC \) ⇒ \( AB = CD \). - Answer:
It involves conditions about angles and lines meeting, unlike the simpler first four postulates. - Answer:
False. A surface has two dimensions (length and breadth). - Answer:
By Axiom 6: Halves of equals are equal. Divide both sides by 2: \( x = y \). - Answer:
Assume two midpoints \( C \) and \( D \). By definition, \( AC = CB \) and \( AD = DB \). Using Axiom 4, \( C \) and \( D \) coincide. Hence, uniqueness. - Answer:
Elements organized geometric knowledge into a logical framework, influencing mathematics for centuries. - Answer:
It is non-intuitive and resembles a theorem. Later, it was found to define non-Euclidean geometries. - Answer:
Consistency means no axiom contradicts another. Euclid’s system avoids paradoxes. - Answer:
Draw circles with radius \( AB \) from \( A \) and \( B \). Their intersection \( C \) forms \( \triangle ABC \). By Postulate 3 and Axiom 1, \( AB = AC = BC \). - Answer:
If \( AB = CD \) and \( CD = EF \), then \( AB = EF \). Applies to line segments, angles, etc. - Answer:
The line joining the centers bisects the common chord at right angles (derived from congruent triangles). - Answer:
Defining them would require more basic terms, leading to circular definitions. - Answer:
Deductive reasoning uses axioms/postulates to derive theorems, ensuring logical validity. - Answer:
By Axiom 4: \( AC + CB \) coincides with \( AB \), hence equal (Example 1 in the text).
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You can also check the related MCQs for this chapter here: MCQ on Introduction to Euclid’s Geometry
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Abdur Rohman is an Electrical Engineer from Charaideo, Assam, who wears multiple hats as a part-time teacher, blogger, entrepreneur, and digital marketer. Passionate about education, he founded The Assam School blog to provide free, comprehensive textbook solutions, MCQs (Multiple Choice Questions), and other academic content for students from Class V to XII.