Chapter 5: Introduction to Euclid’s Geometry – Notes, Practice & Exercise Solutions (Class 9, NCERT/CBSE)
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1-Mark Questions (20)
1) What is meant by an “undefined term” in geometry?
A basic concept accepted without formal definition, e.g., point, line, plane.
2) Name Euclid’s book that systematised geometry.
Elements (13 books).
3) State Euclid’s Postulate 1.
A straight line may be drawn from any one point to any other point.
4) State Euclid’s Postulate 2 in today’s words.
A line segment can be extended indefinitely in both directions to form a line.
5) What does Postulate 3 allow you to construct?
A circle with any centre and any radius.
6) Give the common notion: “If equals are added to equals …”
“… the wholes are equal.”
7) Fill in the blank: “The whole is ____ than the part.”
greater
8) What is another name for the “coordinate plane” in honour of René Descartes?
The Cartesian plane.
9) State Axiom 5.1 used in this chapter.
Given two distinct points, there is a unique line that passes through them.
10) True/False: All right angles are equal.
True (Euclid’s Postulate 4).
11) Which axiom justifies superposition (coinciding figures are equal)?
“Things which coincide with one another are equal to one another.”
12) Define a line segment.
A part of a line bounded by two endpoints.
13) Write the statement of Theorem 5.1 from this chapter.
Two distinct lines cannot have more than one point in common.
14) Which civilisation compiled Elements?
Greek civilisation (Euclid at Alexandria).
15) Fill in: “Things equal to the same thing are _____ to one another.”
equal
16) What is a postulate/axiom?
An assumption accepted as universally true without proof.
17) Give one undefined term required before defining “radius of a circle”.
Point (also circle, distance).
18) Which postulate involves interior angles less than two right angles?
Euclid’s Postulate 5 (parallel postulate form).
19) Name the Greek mathematician credited with the first known proof about a circle’s diameter bisecting it.
Thales.
20) What is a theorem?
A statement proved using axioms, definitions, and previously proved results.
2-Mark Questions (20)
1) Use a common notion to justify: If \(AB=PQ\) and \(PQ=XY\), then \(AB=XY\).
By “things equal to the same thing are equal to one another”, \(AB=XY\).
2) Which postulate allows you to draw a circle with centre \(A\) and radius \(AB\)? State it.
Postulate 3: A circle can be drawn with any centre and any radius.
3) Why is “point” taken as an undefined term?
Any attempt to define it leads to further undefined words (infinite regress); we accept it intuitively.
4) Give a real-life illustration of Axiom (5): “The whole is greater than the part.”
A full pizza (whole) has more quantity than any single slice (part).
5) State whether the following is consistent: “There exist at least three points not on the same line.”
Yes; e.g., vertices of any non-collinear triangle. It does not contradict Euclid’s postulates.
6) Write the Euclidean axiom that allows subtraction of equals from equals.
“If equals are subtracted from equals, the remainders are equal.”
7) Define perpendicular lines (in words).
Two lines that meet to form a right angle (90°).
8) What does Axiom 5.1 conclude about lines through two distinct points \(P\) and \(Q\)?
Exactly one line passes through both \(P\) and \(Q\).
9) Identify the axiom used when we say “overlapping (coinciding) figures are equal.”
Common notion: Things which coincide with one another are equal to one another.
10) Give two examples of magnitudes of different kinds that cannot be compared.
Length of a line vs. area of a rectangle; angle vs. volume of a cube.
11) Write the converse-like use of Axiom (2): If \(a=c\) and \(b=d\), then \(a+b=c+d\). Which axiom is this?
Axiom (2) “If equals are added to equals, the wholes are equal.”
12) Name the postulate used to say a “terminated line can be produced indefinitely”.
Euclid’s Postulate 2.
13) Is “between” an undefined term in Euclid’s system? Briefly comment.
Yes; “between” relies on order on a line and is treated as primitive in Euclidean settings.
14) What is meant by a “consistent” system of axioms?
It is impossible to deduce a statement that contradicts the axioms or earlier theorems.
15) Give a quick proof step: If \(AC=AB+BC\) and \(AB=BC\), show \(AC=2AB\).
Substitute \(BC=AB\): \(AC=AB+AB=2AB\).
16) State Euclid’s Postulate 4 and one consequence.
All right angles are equal. ⇒ Perpendicular is a well-defined concept.
17) Which postulate concerns interior angles less than two right angles implying intersection?
Postulate 5 (parallel postulate style statement).
18) Define “parallel lines”.
Lines in the same plane that do not meet, however far extended.
19) Why can’t there be two different lines through the same two points?
It would violate Axiom 5.1 (uniqueness of line through two points).
20) What is a proposition in Euclid’s Elements?
A statement (problem or theorem) proved using postulates/axioms and prior results.
3-Mark Questions (20)
1) Prove: If \(B\) is between \(A\) and \(C\) on a line, then \(AB+BC=AC\).
By segment addition, \(AC\) coincides with \(AB\) and \(BC\). By the common notion “things which coincide are equal”, \(AB+BC=AC\).
2) Using Postulate 3, construct an equilateral triangle on a given segment \(AB\), and justify it.
Draw circles with centres \(A\) and \(B\), radius \(AB\). Let them meet at \(C\). Then \(AC=AB\) and \(BC=AB\) (radii). Hence \(AB=BC=CA\), so \(\triangle ABC\) is equilateral.
3) State and prove Theorem 5.1: Two distinct lines cannot have more than one common point.
If two distinct lines had two common points \(P,Q\), both lines would pass through \(P\) and \(Q\). By Axiom 5.1, only one line can pass through two distinct points — contradiction. Hence at most one common point.
4) Show that if two circles are equal, their radii are equal.
By definition of equal circles (coinciding magnitudes), corresponding radii coincide; by the axiom “things which coincide are equal”, radii are equal.
5) If a point \(C\) lies between \(A\) and \(B\) with \(AC=BC\), prove \(AC=\\tfrac12 AB\).
Since \(AB=AC+CB\) and \(AC=CB\), \(AB=2AC\\Rightarrow AC=\\tfrac12 AB\).
6) Prove uniqueness of the line through two points without coordinates.
Assume two distinct lines through \(P,Q\). Then both pass through two distinct points \(P,Q\), contradicting Axiom 5.1. Hence unique.
7) Explain why “the whole is greater than the part” is a universal truth.
It applies to any magnitude (length, area, volume, number): a quantity equals its part plus a remainder \(>0\); hence whole \(>\) part, independent of geometry.
8) Using addition axiom, show: If \(AB=CD\) and \(BC=DE\), then \(AB+BC=CD+DE\).
Add equals to equals: \(AB+BC=CD+DE\).
9) Give a geometric interpretation of Postulate 5 with a sketch description.
If a transversal makes interior angles on the same side sum \(<180^\circ\), the two lines meet on that side when extended; e.g., with a line \(t\) crossing lines \(l,m\) with \(\angle\)sum \(<180^\circ\), \(l\) and \(m\) intersect on that side.
10) Show that if \(AC=BD\) on a straight line with \(A,B,C,D\) in order \(A\!-\!B\!-\!C\!-\!D\), then \(AB=CD\).
\(AC=AB+BC\) and \(BD=BC+CD\). Given \(AC=BD\). Subtract equal \(BC\): by Axiom (3) remainders equal ⇒ \(AB=CD\).
11) Prove that a line segment has one and only one midpoint.
Existence: Construct circles with centres \(A\) and \(B\) and equal radii \(>\\tfrac12 AB\). They meet at \(P,Q\). The line \(PQ\) is the perpendicular bisector of \(AB\) and meets \(AB\) at \(M\) with \(AM=MB\). Uniqueness: If \(M,N\) both satisfy \(AM=MB\), the perpendicular bisector would be two different lines through \(A\) and \(B\), impossible. Hence unique midpoint.
12) From Axiom (6) and (7), deduce: If two areas are equal, then their doubles are equal and their halves are equal.
Axiom (6): doubles of equals are equal; Axiom (7): halves of equals are equal.
13) Show that if \(AB=CD\) and \(CD=EF\), then \(AB,CD,EF\) are all equal.
Transitivity via “things equal to the same thing are equal”: \(AB=CD=EF\).
14) Why can’t we compare a line segment to a triangle directly in Euclid’s system?
They are magnitudes of different kinds (length vs. area); only same-kind magnitudes are comparable.
15) Construct a circle through three non-collinear points and explain the axiom/postulate used.
There exists a unique circle through three non-collinear points (circumcircle) using perpendicular bisectors. Uses Postulate 3 (circles) plus uniqueness of perpendicular bisectors.
16) If a straight line intersects two lines such that a pair of alternate interior angles are equal, what can we say in Euclidean geometry?
The lines are parallel (equivalent to Postulate 5).
17) Show: If \(AB>AC\) on the same ray from \(A\), then \(B\) lies beyond \(C\) on \(AC\).
Because “whole greater than part”: \(AB=AC+CB\\Rightarrow CB>0\), so \(B\) lies beyond \(C\) on the same line.
18) Using Postulate 2, argue why a line has no endpoints.
Any terminated line can be extended indefinitely on both sides; hence a (full) line has no endpoints.
19) Give an example where “coinciding implies equality” is used in measurement.
Placing two rulers atop each other; if marks coincide throughout, their lengths are equal.
20) Explain why axioms and postulates are necessary before proving theorems.
They provide foundational truths; without accepted starting points, no deductive proof chain can begin.
Exercise 5.1 – Solutions
1) State True/False with reason:
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) If \(AB=PQ\) and \(PQ=XY\), then \(AB=XY\).
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) If \(AB=PQ\) and \(PQ=XY\), then \(AB=XY\).
(i) False — infinitely many lines pass through one point.
(ii) False — by Axiom 5.1, exactly one line passes through two distinct points.
(iii) True — Euclid’s Postulate 2.
(iv) True — equal circles have equal radii (coincide ⇒ equal).
(v) True — “things equal to the same thing are equal to one another.”
(ii) False — by Axiom 5.1, exactly one line passes through two distinct points.
(iii) True — Euclid’s Postulate 2.
(iv) True — equal circles have equal radii (coincide ⇒ equal).
(v) True — “things equal to the same thing are equal to one another.”
2) Define the following and list prior undefined terms if needed: (i) parallel lines (ii) perpendicular lines (iii) line segment (iv) radius of a circle (v) square.
(i) Parallel lines: Coplanar lines that never meet on extension. (Needs: point, line, plane.)
(ii) Perpendicular lines: Lines meeting to form a right angle. (Needs: line, right angle.)
(iii) Line segment: Part of a line bounded by two endpoints. (Needs: point, line.)
(iv) Radius: Segment joining centre of a circle to a point on the circle. (Needs: point, circle, distance.)
(v) Square: A quadrilateral with four equal sides and four right angles. (Needs: point, line, angle, equality.)
(ii) Perpendicular lines: Lines meeting to form a right angle. (Needs: line, right angle.)
(iii) Line segment: Part of a line bounded by two endpoints. (Needs: point, line.)
(iv) Radius: Segment joining centre of a circle to a point on the circle. (Needs: point, circle, distance.)
(v) Square: A quadrilateral with four equal sides and four right angles. (Needs: point, line, angle, equality.)
3) Consider “postulates”:
(i) Given any two distinct points \(A,B\), there exists a third point \(C\) between \(A\) and \(B\).
(ii) There exist at least three points not on the same line.
Do these contain undefined terms? Are they consistent? Do they follow from Euclid’s postulates?
(i) Given any two distinct points \(A,B\), there exists a third point \(C\) between \(A\) and \(B\).
(ii) There exist at least three points not on the same line.
Do these contain undefined terms? Are they consistent? Do they follow from Euclid’s postulates?
Undefined terms: point, line, “between”.
Consistency: Yes; these hold in the Euclidean plane (no contradiction with Euclid’s postulates).
Do they follow? Not directly; they are independent assumptions about order/existence, not deduced from Euclid’s five postulates.
Consistency: Yes; these hold in the Euclidean plane (no contradiction with Euclid’s postulates).
Do they follow? Not directly; they are independent assumptions about order/existence, not deduced from Euclid’s five postulates.
4) If \(C\) lies between \(A\) and \(B\) and \(AC=BC\), prove \(AC=\\tfrac{1}{2}AB\). (Draw a neat figure.)
Since \(AB=AC+CB\) and \(AC=CB\), \(AB=2AC\\Rightarrow AC=\\tfrac{1}{2}AB\). The symmetric argument gives \(BC=\\tfrac{1}{2}AB\) as well.
5) In Q.4, \(C\) is called the midpoint of \(AB\). Prove that every line segment has one and only one midpoint.
Existence: On segment \(AB\), draw two circles with centres \(A\) and \(B\) and equal radii \(>\\tfrac12 AB\) (Postulate 3). Let them intersect at \(P,Q\). Line \(PQ\) meets \(AB\) at \(M\). By symmetry (or SSS on \(\\triangle APQ\\) and \(\\triangle BPQ\)), \(AM=MB\); hence \(M\) is a midpoint.
Uniqueness: If both \(M,N\) are midpoints, then both lie on the perpendicular bisector of \(AB\). But the perpendicular bisector is a single line; otherwise two different bisectors would pass through the same two points \(A,B\), contradicting Axiom 5.1. Hence the midpoint is unique.
Uniqueness: If both \(M,N\) are midpoints, then both lie on the perpendicular bisector of \(AB\). But the perpendicular bisector is a single line; otherwise two different bisectors would pass through the same two points \(A,B\), contradicting Axiom 5.1. Hence the midpoint is unique.
6) In Fig. 5.10, if \(AC=BD\), prove \(AB=CD\).
On a straight line with points in order \(A\!-\!B\!-\!C\!-\!D\), we have \(AC=AB+BC\) and \(BD=BC+CD\). Given \(AC=BD\). Subtract equals (\(BC\)) from equals (Axiom 3): \(AB=CD\).
7) Why is Axiom 5 (“The whole is greater than the part”) considered a universal truth?
It applies to all magnitudes (length, area, volume, number, etc.) irrespective of geometry. Any whole equals its part plus a positive remainder, so it is strictly greater than the part.
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