| --- |
| source: ncert_exercises |
| topic: geometry |
| class_level: class_9 |
| chapter: ch5 |
| exercise: ex5.1 |
| difficulty: intermediate |
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| Class 9 | Ch5: Introduction to Euclid's Geometry | Exercise 5.1 |
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| 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. |
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| 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. |
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| 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. ✅ |
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| 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. ✅ |
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| 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. ✅ |
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| 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. |
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| 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. ✅ |
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| 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. ✅ |
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| 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. ✅ |
