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source: ncert_exercises
topic: geometry
class_level: class_9
chapter: ch5
exercise: ex5.1
difficulty: intermediate
---
Class 9 | Ch5: Introduction to Euclid's Geometry | Exercise 5.1
Euclid's 5 Postulates:
P1: A straight line can be drawn from any point to any other point.
P2: A terminated line (segment) can be extended indefinitely.
P3: A circle can be drawn with any centre and any radius.
P4: All right angles are equal to one another.
P5: If a line falls on two lines making interior angles on same side less than 180°, the lines meet on that side.
Euclid's Axioms (Common Notions):
A1: Things equal to same thing are equal to each other.
A2: If equals added to equals, wholes are equal.
A3: If equals subtracted from equals, remainders are equal.
A4: Things coinciding with one another are equal.
A5: Whole is greater than part.
Q1. Which statements are true and which false? Give reasons.
(i) Only one line can pass through a single point.
(ii) There are infinite lines through two distinct points.
(iii) A terminated line can be extended indefinitely on both sides.
(iv) If two circles are equal, their radii are equal.
(v) If AB=PQ and PQ=XY, then AB=XY.
Answer:
(i) FALSE. Infinite lines can pass through a single point. ✅
(ii) FALSE. Exactly ONE unique line passes through two distinct points (Postulate 1). ✅
(iii) TRUE. Postulate 2 says terminated line can be produced indefinitely. ✅
(iv) TRUE. Equal circles have same radius. ✅
(v) TRUE. Axiom 1: things equal to same thing are equal to each other. AB=PQ and PQ=XY → AB=XY. ✅
Q2. Give definitions for: (i) Parallel lines (ii) Perpendicular lines (iii) Line segment (iv) Radius of circle (v) Square. State what needs to be defined first.
Answer: Need to know: point, line, ray, angle, plane, circle, quadrilateral.
(i) Parallel lines: Two lines in same plane with no common point. l ∥ m. ✅
(ii) Perpendicular lines: Two lines forming a right angle. p ⊥ q. ✅
(iii) Line segment: Part of a line with two endpoints. ✅
(iv) Radius: Line segment from centre of circle to any point on circle. ✅
(v) Square: Quadrilateral with all four angles = 90° and all four sides equal. ✅
Q3. Consider these postulates: (i) Given any two distinct points A and B, there exists a third point C between A and B. (ii) There exist at least three points not on the same line. Do they contain undefined terms? Are they consistent? Do they follow from Euclid's postulates?
Answer: YES, undefined terms: "point" and "line". They ARE consistent — (i) and (ii) deal with different situations, neither contradicts the other. They do NOT follow directly from Euclid's postulates but are consistent with them. ✅
Q4. If C lies between A and B such that AC=BC, prove that AC = (1/2)AB.
Answer: AC=BC (given). AC+BC=AB (C is between A and B, Axiom 2). AC+AC=AB (substituting BC=AC). 2AC=AB. AC=AB/2=(1/2)AB. ✅ Proved.
Q5. In Q4, C is midpoint of AB. Prove every line segment has one and only one midpoint.
Answer: Suppose C and D are both midpoints of AB. Then AC=CB and AD=DB. AC=AB/2 and AD=AB/2. So AC=AD. But C and D are the same point! Hence midpoint is unique. ✅
Q6. In the figure, if AC=BD, then AB=CD. Explain.
Answer: AC=AB+BC (A,B,C,D in order). BD=BC+CD. Given AC=BD → AB+BC=BC+CD. Subtracting BC from both sides (Axiom 3): AB=CD. ✅
Q7. Why is Axiom 5 considered universal truth but Postulate 5 is not?
Answer: Axiom 5 (whole > part) is true in ALL fields of knowledge — mathematics, science, daily life. Postulate 5 applies only to geometry. Axioms are universal; postulates are specific to a discipline. ✅ |