Chapter 5: Introduction to Euclid's Geometry (NCERT Solutions)
Exercise 5.1
(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) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
(i) False. Through a single point, infinitely many lines can be drawn. Think of the point as the center of a wheel and the lines as its spokes.
(ii) False. According to Euclid's Axiom 5.1, "Given two distinct points, there is a unique line that passes through them." Only one distinct straight line passes exactly through two specific points.
(iii) True. According to Euclid's Postulate 2, "A terminated line can be produced indefinitely." In modern geometry, a terminated line is what we call a line segment, and it can be extended in both directions to form a line.
(iv) True. The area of a circle depends directly on its radius (πr²). If two circles are completely equal (i.e. they superimpose on each other perfectly, thereby having equal areas), their boundaries will perfectly coincide, proving their radii must be exactly the same length.
(v) True. According to Euclid's first axiom, "Things which are equal to the same thing are equal to one another." Since both AB and XY are equal to PQ, AB must be equal to XY.
(i) parallel lines
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square
Yes, there are foundational terms like strictly undefined terms: point, line, plane, and other defined terms like angle, circle, and intersect which we build upon.
(i) Parallel lines: If the perpendicular distance between two lines is always constant, then they are parallel. In other words, two straight lines in the same plane that never intersect, no matter how far they are extended. (We must define 'line', 'intersect' and 'plane' first).
(ii) Perpendicular lines: Two intersecting lines are perpendicular if they form a right angle (90°) at their point of intersection. (We must define 'intersect', 'line', and 'right angle').
(iii) Line segment: A part of a line with two distinct end points is a line segment. (We must define 'point' and 'line').
(iv) Radius of a circle: The distance from the center of the circle to any point on its circumference. (We must define 'circle', 'center', and 'point').
(v) Square: A quadrilateral with all four sides equal in length and all interior angles equal to 90° (right angles). (We must define 'quadrilateral', 'right angle', and 'sides').
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid's postulates? Explain.
Undefined terms: Yes, 'point' and 'line' are undefined terms present in these postulates.
Consistency: Yes, these postulates are consistent because they refer to two
completely different, non-contradictory situations.
Postulate (i) talks of points lying strictly on a line segment.
Postulate (ii) confirms the dimensional nature of a plane, stating points don't always fall
collinear.
Relation to Euclid's Postulates: No, these postulates do not directly follow from Euclid's five Postulates. However, they follow logically from Euclid's Axiom 5.1 ("Given two distinct points, there is a unique line that passes through them").
By geometry given: Point C lies perfectly between A and B, so A, C, B are collinear points.
Therefore, we can establish: AC + CB = AB ...(Eq 1)
It is given that: AC = BC.
Let us substitute BC with AC in equation (1):
AC + AC = AB
2AC = AB
AC = ½ AB (proved).
(Draw a straight line segment labeling points A and B at the ends, and point C exactly in the center).
Let's use the method of contradiction. Let us assume the line segment AB has two midpoints, C and D.
Since C is a midpoint of AB: AC = ½ AB ...(Eq 1)
Since D is a midpoint of AB: AD = ½ AB ...(Eq 2)
According to Euclid's axiom: "Things which are equal to the same thing are equal to one
another."
Therefore, AC = AD.
However, this is physically impossible unless point C coincides precisely with point D, because
AC and AD lie on the same line segment extending from the exact same starting point A.
Thus, C and D must be the identical point.
Therefore, a line segment can have one and only one midpoint.
From the figure, points A, B, C, D are collinear in that order.
We are given: AC = BD
From the line segment construction, we can clearly see:
AC = AB + BC
BD = BC + CD
Now substitute these expressions into the given equation:
AB + BC = BC + CD
According to Euclid's Axiom: "If equals are subtracted from equals, the remainders are
equal."
Let us subtract BC from both sides:
AB + BC - BC = BC + CD - BC
AB = CD. (Proved).
Euclid's Axiom 5 states: "The whole is strictly greater than the part."
This is considered a universal truth because it vividly applies to any measurable or quantifiable entity anywhere in the universe. Whether you're dividing an apple into segments, assigning portions of land, or partitioning time, a partial piece (a part) of an object can never exceed or equal the size of the entire, intact, original item (the whole). It is a universally observable fact.
Exercise 5.2
Euclid's fifth postulate is quite long. A simpler way to understand it is via Playfair's Axiom:
"For every straight line 'l' and for every point 'P' not lying on 'l', there exists a unique straight line 'm' passing through 'P' which is strictly parallel to 'l'."
Another intuitive way to write it: "If a line intersects two distinct straight lines causing the interior angles on the same side to sum to less than 180°, then those two lines, if extended forever, will eventually crash into one another on that exact side."
Yes, Euclid's fifth postulate implies the definitive existence of parallel lines.
According to the Postulate 5: If a straight line falling on two straight lines creates consecutive interior angles whose sum is strictly less than 180° (two right angles), then the two straight lines will meet on that side when extended.
Conversely: If the sum of the interior angles on the same side is exactly equal to 180° (∠1 + ∠2 = 180°), then the lines will not meet on that side. Furthermore, the sum on the other side will logically also be 180°. Therefore, the lines will never meet on either side, no matter how far they are produced.
Two coplanar straight lines that never meet are called parallel lines. Thus, the condition of the interior angles summing to exactly 180° ensures the existence of parallel lines.
Chapter 5: Introduction to Euclid's Geometry (Practice Questions)
RD Sharma / Extra Practice Questions
The altars used for household rituals were primarily squares and circles.
The three primarily undefined terms acting as the foundational building blocks of Euclidean
Geometry are:
1. Point
2. Line
3. Plane (or Surface)
Axioms are generalized, self-evident universal truths that are assumed to be
true universally in all branches of mathematics (e.g., "The whole is greater than the
part").
Postulates are specifically self-evident assumptions that are strictly specific
to geometry alone (e.g., "A straight line may be drawn from any one point to any other point").
Let's say we have two identical glass beakers containing exact equal amounts of water: Beaker A =
500ml and Beaker B = 500ml.
Now, we add an equal amount of water (100ml) to both.
New volume in A = 500 + 100 = 600ml.
New volume in B = 500 + 100 = 600ml.
The whole quantities (600ml) remain firmly equal to each other.
Postulate 1: A straight line may be drawn connecting any one point to any other distinct point. (This implicitly guarantees that given two distinct points, there is a unique line passing through them).
A point has zero dimensions (0). It signifies an exact position but has no length, width, or thickness.
A Theorem is a mathematical statement or proposition that is not self-evident
but must be definitively proved via logical reasoning, using previously established axioms,
postulates, and definitions.
In contrast, an Axiom requires no proof. It is accepted universally as
self-evidently true.
Euclid defined that the edges or boundaries of a flat surface are lines (or curves).
Assume an angle ∠ABC = 60°.
Let line BP bisect ∠ABC. So, ∠ABP = 30°.
Let another line BQ also act as a formal bisector of a identical separate 60° angle,
∠XYZ. So ∠XBQ = 30°.
Since ∠ABP = ½ (60°) and ∠XBQ = ½ (60°), then logically
∠ABP = ∠XBQ (30° = 30°).
Given: AB = BC ...(1)
And given: BX = BY ...(2)
According to Euclid's Axiom 3: "If equals are subtracted from equals, the remainders are
equal."
Subtract equation (2) from equation (1):
AB - BX = BC - BY
Therefore, AX = CY. (Proved).
According to Euclid's first axiom: "Things which are equal to the same thing are equal to
one another."
Since x = z, we can directly substitute z in place of x in the original equation.
Substitute x with z in (x + y = 10).
Therefore, z + y = 10.
Postulate 4: "All right angles are perfectly equal to one another." (This means every 90° angle, regardless of its position or the lengths of the lines forming it, is congruent to any other 90° angle).
Only one valid unique line can pass linearly through two distinct points. This is an application of Euclid's first Postulate/Axiom 5.1.
According to Euclid's Axiom 4: "Things which coincide with one another are equal to one
another."
If P lies directly on the segment AB between A and B, the combination of segment AP and segment
PB perfectly coincides with the whole segment AB.
Therefore, AP + PB = AB.
Postulate 3: "A circle can be formally drawn with any central point and any designated radius."
Chapter 5: Introduction to Euclid's Geometry (Summary & Definitions)
1. Euclid's Basic Definitions
- Point: A point is that which has no part (no dimensions).
- Line: A line is breathless length.
- Ends of a Line: The ends of a line are points.
- Straight Line: A straight line is a line which lies evenly with the points on itself.
- Surface: A surface is that which has length and breadth only.
- Edges of a Surface: The edges of a surface are lines.
- Plane Surface: A plane surface is a surface which lies evenly with the straight lines on itself.
- Modern Note: In modern geometry, point, line, and plane are considered strictly undefined terms because defining them systematically requires an infinite regression of prior definitions.
2. Euclid's Seven Axioms
(Axioms are universally accepted self-evident truths applicable to all branches of mathematics)
- Things which are logically equal to the exact same thing are also equal to one another. (If a = c and b = c, then a = b).
- If equals are explicitly added to equals, the wholes are equal. (If a = b, then a + c = b + c).
- If equals are visibly subtracted from equals, the remainders are equal. (If a = b, then a - c = b - c).
- Things which coincide completely with one another are definitively equal to one another. (Crucial for geometrical superposition).
- The whole is absolutely greater than the part.
- Things which are double of the same thing are equal to one another. (If a = 2c and b = 2c, then a = b).
- Things which are halves of the same thing are equal to one another. (If a = c/2 and b = c/2, then a = b).
3. Euclid's Five Postulates
(Postulates are self-evident assumptions specifically structured for geometry alone)
- Postulate 1: A straight line may be flawlessly drawn from any one point to any
other point.
(Axiom 5.1 extension: Given two distinct points, there is a singular unique line that strictly passes through them). - Postulate 2: A legitimately terminated line (line segment) can be visibly produced indefinitely in both directions to form a line.
- Postulate 3: A perfect circle can be analytically drawn with any central point and any designated radius.
- Postulate 4: All structured right angles exactly equal one another mathematically (90°).
- Postulate 5: If a straight line falling securely on two straight lines makes the absolute inner angles on the identical side of it taken jointly less than two right angles (180°), then the two straight lines, if structurally produced ad infinitum, intersect on precisely that side on which the total of angles is visibly less than 180°.
4. Equivalent Versions of Postulate 5
- Playfair's Axiom: "For every line 'l' and for every defined point 'P' not explicitly lying on 'l', there exists a unique line 'm' formally passing through 'P' such that 'm' is visibly parallel to 'l'."
- "Two distinct intersecting mathematical lines definitively cannot be parallel to the identical line."
- A complete system of logically deduced statements proving theorems based directly upon axioms/postulates establishes the backbone of deductive Euclidean Geometry.