CBSE Class 9 Mathematics Euclids Geometry Worksheet Set 02

1. Euclid’s Definitions, Axioms and Postulates
2. Equivalent Versions of Euclid’s Fifth Postulate
The Greeks developed geometry is a systematic manner Euclid (300 B.C.) a greek mathematician, father of geometry introduced the method of proving mathematical results by using deductive logical reasoning and the previously proved result. The Geometry of plane figure is known as "Euclidean Geometry". 
Axioms: The basic facts which are taken for granted without proof are called axioms some Euclid's axioms are:
(i) Things which are equal to the same thing are equal to one another. i.e. a = b, b = c⇒a = c
(ii) If equals are added to equals, the wholes are equal i.e. a = b ⇒ a + c = b + c
(iii) If equals are subtracted from equals, the remainders are equal i.e. a = b⇒a - c = b - c
(iv) Things which coincide with one another are equal to one another.
(v) The whole is greater than the part.
Postulates: Axioms are the general statements, postulates are the axioms relating to a particular field.
Educlid's five postulates are.
(i) A straight line may be drawn from any one point to any other point.
(ii) A terminated line can be produced indefinitely.
(iii) A circle can be drawn with any centre and any radius.
(iv) All right angles are equal to one another.
(v) If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which the angles are less than two right angles.
Statements: A sentence which is either true or false but not both, is called a statement.
eg. (i) 4+9=6 If is a false sentence, so it is a statement.
(ii) Sajnay is tall. This is not a statement because he may be tall for certain persons and may not be taller for others.
Theorems: A statement that requires a proof is called a theorem.
eg. (i) The sum of the angles of triangle is 180o .
(ii) The angles opposite to equal sides of a triangles are equal.
Corollary - Result deduced from a theorem is called its corollary.

Question. The three steps from solids to points are:
(a) Solids - surfaces - lines - points
(b) Solids - lines - surfaces - points
(c) Lines - points - surfaces - solids
(d) Lines - surfaces - points – solids

Answer: A

Question. The number of dimension, a point has:
(a) 0
(b) 1
(c) 2
(d) 3

Answer: A

Question. The total number of propositions in the Elements are:
(a) 465
(b) 460
(c) 13
(d) 55

Answer: A

Question. The number of dimensions, a solid has:
(a) 1
(b) 2
(c) 3
(d) 0

Answer: C

Question. Boundaries of solids are:
(a) surfaces
(b) curves
(c) lines
(d) points

Answer: A

Question. Boundaries of surfaces are:
(a) surfaces
(b) curves
(c) lines
(d) points

Answer: B

Question. In ancient India, the shapes of altars used for house hold rituals were:
(a) Squares and circles
(b) Triangles and rectangles
(c) Trapeziums and pyramids
(d) Rectangles and squares

Answer: A

Question. A pyramid is a solid figure, the base of which is
(a) only a triangle
(b) only a square
(c) only a rectangle
(d) any polygon

Answer: D

Question. The side faces of a pyramid are:
(a) Triangles
(b) Squares
(c) Polygons
(d) Trapeziums

Answer: A

Question. It is known that if x + y = 10 then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is:
(a) First Axiom
(b) Second Axiom
(c) Third Axiom
(d) Fourth Axiom

Answer: B

Question. In Indus Valley Civilisation (about 3000 B.C.), the bricks used for construction work
were having dimensions in the ratio
(a) 1 : 3 : 4
(b) 4 : 2 : 1
(c) 4 : 4 : 1
(d) 4 : 3 : 2

Answer: B

Question. Euclid divided his famous treatise “The Elements” into:
(a) 13 chapters
(b) 12 chapters
(c) 11 chapters
(d) 9 chapters

Answer: A

Question. The number of interwoven isosceles triangles in Sriyantra (in the Atharva Veda) is:
(a) Seven
(b) Eight
(c) Nine
(d) Eleven

Answer: C

Question. In Ancient India, Altars with combination of shapes like rectangles, triangles and trapeziums were used for:
(a) Public worship
(b) Household rituals
(c) Both A and B
(d) None of A, B, C

Answer: A

Question. Euclid stated that all right angles are equal to each other in the form of
(a) an axiom
(b) a definition
(c) a postulate
(d) a proof

Answer: C

Question. Euclid belongs to the country:
(a) Babylonia
(b) Egypt
(c) Greece
(d) India

Answer: C

Question. Greek’s emphasised on:
(a) Inductive reasoning
(b) Deductive reasoning
(c) Both A and B
(d) Practical use of geometry

Answer: B

Question. Thales belongs to the country:
(a) Babylonia
(b) Egypt
(c) Greece
(d) Rome

Answer: C

Question. The number of dimensions, a surface has:
(a) 1
(b) 2
(c) 3
(d) 0

Answer: B

Question. Which of the following needs a proof?
(a) Theorem
(b) Axiom
(c) Definition
(d) Postulate

Answer: A

Question. ‘Lines are parallel if they do not intersect’ is stated in the form of
(a) an axiom
(b) a definition
(c) a postulate
(d) a proof

Answer: B

Question. Pythagoras was a student of:
(a) Thales
(b) Euclid
(c) Both A and B
(d) Archimedes

Answer: A

Question. The boundaries of the solids are curves.
Answer. The given statement is false because boundaries of solids are surfaces.

Question. The statements that are proved are called axioms.
Answer. The given statement is false because the statement that are proved are called theorems.

Question. The things which are double of the same thing are equal to one another.
Answer. True Since, it is one of the Euclid’s axioms. Some of Euclid’s axiom:
(1) Things which are equal to the same thing are equal to one another.
(2) If equals are added to equals, the wholes are equal.
(3) If equals are subtracted from equals, the remainder are equal.
(4) Things which coincide with one another are equal to one another.
(5) The whole is greater than the part.
(6) Things which are double of the same things are equal to one another.
(7) Thing which are halves of the same things are equal to one another.

Question. Euclidean geometry is valid only for curved surfaces.
Answer. The given statement is false because Euclidean geometry is valid only for the figures in the plane.

Question. Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.
Answer. The given statement is true because these geometries are different from Euclidean geometry called non-Euclidean geometry.

Question. If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C.
Answer. The given statement is true because it is one of Euclid’s axiom.

Question. “For every line l and for every point P not lying on a given line l, there exists a unique line m passing through P and parallel to l” is known as Play fair’s axiom.
Answer. The given statement is true, because it is an equivalent version of Euclid’s fifth postulate.

Question. The edges of a surface are curves.
Answer. The given statement is false because the edges of surfaces are line.

Question. Two distinct intersecting lines cannot be parallel to the same line.
Answer. The given statement is true, because it is an equivalent version of Euclid’s fifth postulate.

Question. Two salesmen make equal sales during the month of August. In September, each salesman doubles his sale of the month of August. Compare their sales in September.
Answer. Let the sales of two salesmen in the month of August be x and y. As, they make equal sale during the month of August, x = y. In September, each salesman double his sale of the month of August, So 2x = 2y.
Now, by Euclid’s axiom, thing which are double of the same things are equal to one another.
Hence, we can say that in the month of September also, two salesmen make equal sales.

Question. Look at the figure, and show that
Length AH > sum of lengths of AB + BC + CD.

Answer. We see the AB, BC and CD are parts of line.
Now, AB + BC + CD = AD                                                         …(1)
By Euclid’s axiom 5, the whole is greater than the part, so AH > AD
i.e., Length AH> sum of length of AB + BC + CD                [Using (1)]

Question. It is known that x + y = 10 and that x = z. Show that z + y= 10?
Answer. It is known that x + y = 10 and that x = z.
∴ x + y = z + y                [∵By Euclid’s axiom 2, if equals of are added to equals, the wholes are equal]
⇒ 10 = y + z                   [Using (1), x + y = 10]
Hence, z + y = 10.

Question. Study the following statement:
“Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two
intersecting lines l and m and the line n in the above statement.]
Answer. Two intersecting lines cannot be both perpendicular to the same line because if two lines l and m are perpendicular to the same line n, then l and m must be parallel. The given statement is not an equivalent version of Euclid’s fifth postulate. 

Question. Read the following two statements which are taken as axioms:
(i) If two lines intersect each other, then the vertically opposite angles are not equal.
(ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°. Is this system of axioms consistent? Justify your answer.
Answer. The given system of axioms is not consistent because if a ray stands on a line and the sum of two adjacent angles so formed is equal to 180o, then for two lines which intersect each other, the vertically opposite angles becomes equal.

Question. Read the following statement:
An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are 60° each. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in a equilateral triangle?
Answer. The term need to be defined are:
Polygon: A simple closed figure made up of three or more line segments.
Line segment: Part of a line with two end points.
Line: Undefined term.
Point: Undefined term.
Angle: A figure formed by two rays with a common initial point.
Acute angle: Angle whose measure is between 0o and 90o.
Undefined terms used are: Line, part Two line segments are equal to third line segment Therefore, all three sides of an equilateral triangle are equal All its angle are 60o each. Therefore, all angles are equal (by Euclid’s first axiom, things which are equal to same things are equal to one another.)
Hence, we can say that all sides and all angles are equal in an equilateral triangle.

Question. Read the following axioms:
(i) Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equal.
(iii) Things which are double of the same thing are equal to one another.
Check whether the given system of axioms is consistent or inconsistent.
Answer. The given system of axioms is consistent because (i), (ii) and (iii) are Euclid’s axiom.

Question. Read the following statements which are taken as axioms:
(i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
(ii) If a transversal intersects two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent? Justify your answer.
Answer. No, this system of axioms is not consistent because if a transversal intersects two parallel lines and if corresponding angles are not equal, then alternate interior angles cannot be equal.

Introduction to Euclid's Geometry

1 Mark Questions

Question 1. A terminated line can be produced
(a) definitely
(b) indefinitely
(c) both A & B
(d) None
Answer: (b) indefinitely
In simple words: According to Euclid's second postulate, a line segment (or terminated line) can be extended endlessly in both directions.

Exam Tip: Euclid referred to a line segment as a "terminated line"; remember that it can be extended without limit on either side.

Question 2. Two lines are said to be coincident if they have only one common Point. (T/F)
Answer: False. If two lines share exactly one point in common, they are defined as intersecting lines. In contrast, coincident lines overlap each other completely and therefore share an infinite number of common points.
In simple words: Lines that cross at just one point are intersecting lines, not coincident lines.

Exam Tip: Do not confuse "intersecting lines" (one common point) with "coincident lines" (every point is shared).

Question 3. Which of the following has two dimension ?
(a) solid
(b) surface
(c) line
(d) point
Answer: (b) surface
In simple words: A solid has three dimensions, a surface has two (length and breadth), a line has one, and a point has zero.

Exam Tip: Remember the dimension hierarchy: solid (3) → surface (2) → line (1) → point (0).

Question 4. The Euclid's geometry was developed by
(a) Thales
(b) Pythagoras
(c) Egypt
(d) Euclid
Answer: (d) Euclid
In simple words: The system of plane geometry was organized and compiled by the Greek mathematician Euclid in his famous treatise "Elements".

Exam Tip: Euclid collected all known mathematical knowledge of his era into 13 books known as "The Elements".

Question 5. A solid has
(a) one dimension
(b) two dimension
(c) three dimension
(d) none
Answer: (c) three dimension
In simple words: Any solid object takes up space with length, width, and height, so it has three dimensions.

Exam Tip: Solids possess shape, size, and position, and their boundaries are surfaces.

Question 6. In Euclid's axioms, If equals are added to equals, the wholes are
(a) unequal
(b) equal
(c) many or many not be equal
(d) none of these
Answer: (b) equal
In simple words: If you start with two equal quantities and add the same amount to both, the final totals remain equal.

Exam Tip: This statement is Euclid's second axiom; memorizing the exact wording of all seven common axioms ensures quick marks.

Question 7. The edges of a surface are
(a) points
(b) length
(c) lines
(d) none
Answer: (c) lines
In simple words: The outer boundaries that enclose any surface are lines.

Exam Tip: Keep in mind Euclid's definitions: surfaces are bounded by lines, and lines end in points.

Question 8. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together their sum is less than the sum of two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is :
(a) < 180º
(b) > 180º
(c) ≤ 180º
(d) ≥ 180º
Answer: (a) < 180º
In simple words: Two lines will tilt toward each other and eventually cross on whichever side the interior angles add up to less than 180 degrees.

Exam Tip: "Two right angles" equals \( 2 \times 90^\circ = 180^\circ \); the lines always meet on the side where the angle sum is strictly less than \( 180^\circ \).

Question 9. Two distinct intersecting lines cannot be parallel to the
(a) same line
(b) distinct line
(c) both a & b
(d) none
Answer: (a) same line
In simple words: If two different lines cross each other, they cannot both be parallel to the exact same straight line.

Exam Tip: This principle is an equivalent form of Playfair's axiom, which derives from Euclid's fifth postulate.

Question 10. For all positive integer n, 2n + 1 is
(a) even
(b) odd
(c) prime
(d) none
Answer: (b) odd
In simple words: Since 2n is always an even number, adding 1 makes it an odd number every single time.

Exam Tip: Any integer multiplied by 2 yields an even number, and adding 1 to any even integer guarantees an odd result.

Question 11. If a point C lies between two points A and B such that AC = BC, then AC =
(a) AB
(b) BC
(c) \( \frac{\text{AB}}{2} \)
(d) None
Answer: (c) \( \frac{\text{AB}}{2} \)
In simple words: Because C sits exactly in the middle between A and B, the segment AC is half the total length of AB.

Exam Tip: Point C is the midpoint of segment AB; write \( \text{AC} + \text{BC} = \text{AB} \) and substitute \( \text{BC} = \text{AC} \) to show \( 2\text{AC} = \text{AB} \).

Question 12. A mathematical statement whose truth has been established (or proved) in called
(a) Theorem
(b) Deductive reasoning
(c) Conjecture
(d) None of these
Answer: (a) Theorem
In simple words: A statement that has been proven true using logical steps is called a theorem.

Exam Tip: Contrast axioms/postulates (self-evident assumptions taken without proof) with theorems (propositions proved using axioms and definitions).

Question 13. A point is that which has
(a) one part
(b) two part
(c) no part
(d) none
Answer: (c) no part
In simple words: By Euclid's first definition, a point has no size, area, or parts.

Exam Tip: Memorize Euclid's Definition 1 verbatim: "A point is that which has no part."

Question 14. Two distinct lines cannot have more than
(a) one point in common
(b) two points in common
(c) three
(d) none
Answer: (a) one point in common
In simple words: Two straight lines can cross each other at only one single point.

Exam Tip: If two lines shared more than one point, they would coincide and form the same line, violating Euclid's axiom that a unique line passes through two points.

Question 15. Only one line can pass through ____distinct points.
(a) single
(b) two
(c) three
(d) none
Answer: (b) two
In simple words: There is only one unique straight line that can connect two given separate points.

Exam Tip: Axiom 5.1 states: "Given two distinct points, there is a unique line that passes through them."

Question 16. If A, B and C are three points on a line, and B lies between A and C then
(a) AC = AB + AC
(b) AC = AB + BC
(c) AB = AB + AC
(d) BC = AC + AB
Answer: (b) AC = AB + BC
In simple words: The total length from A to C is made by adding the distance from A to B and the distance from B to C.

Exam Tip: Euclid's Axiom 4 states that things coinciding with one another are equal; here the segment AC coincides with AB placed next to BC.

Question 17. There are infinite number of lines which pass through
(a) two distinct points
(b) a single point
(c) three points
(d) none
Answer: (b) a single point
In simple words: You can draw countless straight lines in every possible direction through one solitary point.

Exam Tip: Contrast this with two points: infinitely many lines pass through one point, but only one line passes through two distinct points.

Question 18. A circle can be drawn with
(a) any centre & any radius
(b) one centre & one radius
(c) any centre & one radius
(d) one centre & any radius
Answer: (a) any centre & any radius
In simple words: Euclid's third postulate allows a circle to be constructed around any chosen center point using any desired radius length.

Exam Tip: This directly quotes Postulate 3: "A circle can be drawn with any centre and any radius."

Question 19. If two circles are equal, then their radii are
(a) unequal
(b) may be equal
(c) equal
(d) none
Answer: (c) equal
In simple words: When two circles have the exact same size and area, the distance from their centers to their edges must be identical.

Exam Tip: Superimposing one circle onto an equal circle causes their boundaries and centers to coincide, making their radii equal by Axiom 4.

Question 20. Two distinct lines intersect at more than one points (T/F)
Answer: False. Two separate straight lines can meet at a maximum of one single point. If they had two or more points in common, they would be the exact same line.
In simple words: Two straight lines never cross each other more than once.

Exam Tip: State that two points determine a unique line, so intersecting at two points is geometrically impossible for straight lines.

Question 21. In fig. if AC = BD, then
(a) AB = CD
(b) AC = CD
(c) BC = CD
(d) AB = BC
Answer: (a) AB = CD
In simple words: Write AC as AB + BC and BD as BC + CD. Subtracting the shared middle part BC from both leaves AB equal to CD.

Exam Tip: Apply Euclid's third axiom: "If equals are subtracted from equals, the remainders are equal," by taking BC away from both sides.

Question 22. Two distinct intersecting lines cannot be intersecting to the:
(a) Parallel line
(b) Distinct line
(c) both a & b
(d) None
Answer: (d) None
In simple words: Two lines that cross can freely cross other lines as well, so none of the options describes an impossible intersection.

Exam Tip: Read carefully: intersecting lines cannot be *parallel* to the same line, but they can certainly intersect other lines.

Question 23. The kind of reasoning we have used to solve the puzzle is
(a) Mathematical Reasoning
(b) Deductive Reasoning
(c) Both a & b
(d) Conjecture
Answer: (b) Deductive Reasoning
In simple words: Working out conclusions step-by-step from given rules using pure logic is called deductive reasoning.

Exam Tip: Deductive reasoning derives specific conclusions from established general statements or premises.

Question 24. In Euclid geometry the undefined terms are
(a) a point, a line
(b) a plane surface
(c) both a & b
(d) none of these
Answer: (c) both a & b
In simple words: Modern geometry treats point, line, and plane surface as basic starting ideas that cannot be formally defined without circular descriptions.

Exam Tip: Euclid gave descriptive definitions for points, lines, and planes, but modern mathematicians classify all three as strictly undefined terms.

Question 25. Consider the following pattern of numbers called the Pascal's Triangle :

What can you conjecture about the sum of the numbers in Lines 7 and 9? What about the sum of the numbers in Line 25? Do you see a pattern? Make a guess about a formula for the sum of the numbers in line n+1.
(a) \( 2^6, 2^8, 2^{24}, 2^n \)
(b) \( 2^7, 2^9, 2^{25}, 2^{n+1} \)
(c) \( 2^6, 2^8, 2^{25}, 2^n \)
(d) None
Answer: (a) \( 2^6, 2^8, 2^{24}, 2^n \)
In simple words: The sum of line k equals 2 to the power (k - 1). Therefore, line 7 gives \( 2^6 \), line 9 gives \( 2^8 \), line 25 gives \( 2^{24} \), and line n + 1 gives \( 2^n \).

Exam Tip: Look at the exponent pattern: for line \( k \), the sum is \( 2^{k-1} \); substituting \( k = n+1 \) produces \( 2^{(n+1)-1} = 2^n \).

Question 26. Which of the following statement is true.
(a) Only one line can pass through a single point.
(b) A terminating line can produced indefinitely on both the sides.
(c) If two circle are unequal then their circumference are equal.
(d) There are an infinite number of lines which pass through two distinct points.
Answer: (b) A terminating line can produced indefinitely on both the sides.
In simple words: A line segment can be extended endlessly in both directions, which is Euclid's second postulate.

Exam Tip: Statements (a), (c), and (d) are all mathematically false: infinitely many lines pass through one point, and only one passes through two points.

Question 27. Which of the following has three dimension?
(a) Solid
(b) Surface
(c) Line
(d) Point
Answer: (a) Solid
In simple words: A solid object occupies three dimensions: length, breadth, and depth.

Exam Tip: Recall that three-dimensional figures are solids, two-dimensional are planes/surfaces, one-dimensional are curves/lines, and zero-dimensional are points.

Question 28. Will a ray AB be the same as BA? (T/F)
Answer: False. Ray AB originates at point A and extends endlessly past point B, whereas ray BA originates at point B and extends endlessly past point A. Because their initial starting points and directions are completely different, ray AB and ray BA are not identical.
In simple words: They start at different points and travel in opposite directions, so they are two distinct rays.

Exam Tip: For a ray, the order of letters indicates its direction: the first letter is always the fixed origin point.

Question 29. Three or more lines are said to be concurrent if their one _____ lies on them.
Answer: point (or common point). Three or more lines in a plane are called concurrent lines if they all pass through one common point, which is known as their point of concurrency.
In simple words: Lines are concurrent when they all cross through the exact same single point.

Exam Tip: The shared intersection point of concurrent lines is called the "point of concurrency".

Question 30. Given two distinct points, there is a ____ line that passes through them.
Answer: unique (or one and only one). According to Euclid's axiom, exactly one unique straight line connects two separate points.
In simple words: Only one single straight line can pass through two different points.

Exam Tip: "Unique" is the standard keyword examiners look for in Axiom 5.1.

Question 31. A line contains ____ points on it.
Answer: infinitely many. A straight line is an unbroken, continuous collection composed of an infinite number of points.
In simple words: Every line is made up of endless points packed together.

Exam Tip: While two points determine a line, the line itself contains infinitely many points.

Question 32. A point is that which has..... part.
Answer: no. By Euclid's first definition in Book 1 of Elements, a point has zero magnitude and therefore has no part.
In simple words: A point is completely indivisible and has no size.

Exam Tip: Complete this sentence using the word "no" to match Euclid's opening definition.

Question 33. Look at the following pattern :
\( 1^2 = 1 \)
\( 11^2 = 121 \)
\( 111^2 = 12321 \)
Make a conjecture about each of the following :
\( 111111^2 = \)
\( 1111111^2 = \)
(a) 123454321, 12345654321
(b) 12345654321, 12345654321
(c) 12345654321, 1234567654321
(d) None of these
Answer: (c) 12345654321, 1234567654321
In simple words: The digits count up to the total count of 1s in the number and then count back down to 1. Six ones peak at 6, and seven ones peak at 7.

Exam Tip: For any repunit \( (11\ldots1)^2 \) with \( k \) ones (up to \( k = 9 \)), the square is a palindrome rising from 1 to \( k \) and falling back to 1.

Question 34. The main logical tool used in establishing the truth of unambiguous statements is
(a) Theorem
(b) Conjecture
(c) both a and b
(d) Deductive reasoning
Answer: (d) Deductive reasoning
In simple words: Mathematics relies on deductive reasoning to prove that clear statements are undoubtedly true.

Exam Tip: Deductive logic is the core methodology that turns conjectures into established theorems.

Question 35. Euclid's axioms are
(i) If equals are subtracted from equals, the remainders are equal.
(ii) The whole is greater than the part.
(iii) Things which are double of the same things are equal to one another.
(iv) Things which are halves of the same things are equal to one another.
(a) (i) & (iii)
(b) (i), (ii) & (iv)
(c) (i) & (iv)
(d) (i) & (ii)
Answer: (b) (i), (ii) & (iv)
In simple words: All four statements are genuine axioms formulated by Euclid; option (b) contains the largest collection of these valid axioms.

Exam Tip: Keep in mind that statements (i) through (iv) are all included in Euclid's list of seven common notions.

3 Mark Questions

Question 36. If A, B and C are three points on a line, and B lies between A and C then prove that AB + BC = AC.
Fig.
Answer: Consider the line segment AC on which point B is positioned between A and C.
The segment AC can be seen as two adjacent segments, AB and BC, placed end to end.
When segment AB and segment BC are combined, their total length coincides completely with the entire line segment AC.
According to Euclid's Axiom 4: "Things which coincide with one another are equal to one another."
Because the combination of AB and BC coincides with AC, it follows directly that:
\[ \text{AB} + \text{BC} = \text{AC} \]
Hence proved.
In simple words: Putting together segments AB and BC covers segment AC exactly, and things that overlap completely are equal.

Exam Tip: State Euclid's fourth axiom ("Things which coincide with one another are equal to one another") explicitly to justify the proof.

Question 37. If a point C lies between two points A and B such that AC = BC, then prove that \( \text{AC} = \frac{1}{2}\text{AB} \). Explain by drawing the figure.
Answer: We are given that point C lies between points A and B on a line, with \( \text{AC} = \text{BC} \).
From the geometry of the line segment, the total length is the sum of its two parts:
\[ \text{AC} + \text{BC} = \text{AB} \]
Since it is given that \( \text{BC} = \text{AC} \), substitute AC in place of BC:
\[ \text{AC} + \text{AC} = \text{AB} \]
\[ 2\text{AC} = \text{AB} \]
By Euclid's axiom, "Things which are halves of the same things are equal to one another." Dividing both sides by 2 gives:
\[ \text{AC} = \frac{1}{2}\text{AB} \]
Hence proved.
In simple words: Since AC and BC are identical, two ACs equal the whole line AB, meaning one AC is half of AB.

Exam Tip: Draw the line clearly showing C positioned centrally between A and B, and cite the axiom regarding halves of equals.

Question 38. In Fig., if AB = BC, and AC = BD then prove that AB = CD.
Fig.
Answer: From the given figure, points A, B, C, and D lie in order along a straight line.
We are given:
\[ \text{AC} = \text{BD} \quad \text{--- (1)} \]
Notice that segment AC is composed of AB and BC:
\[ \text{AC} = \text{AB} + \text{BC} \quad \text{--- (2)} \]
Similarly, segment BD is composed of BC and CD:
\[ \text{BD} = \text{BC} + \text{CD} \quad \text{--- (3)} \]
Substitute equations (2) and (3) into equation (1):
\[ \text{AB} + \text{BC} = \text{BC} + \text{CD} \]
According to Euclid's Axiom 3: "If equals are subtracted from equals, the remainders are equal."
Subtracting BC from both sides of the equation yields:
\[ \text{AB} + \text{BC} - \text{BC} = \text{BC} + \text{CD} - \text{BC} \]
\[ \text{AB} = \text{CD} \]
Hence proved.
In simple words: Both AC and BD include the middle segment BC. When you subtract BC from both equal lengths, the remaining ends AB and CD must be equal.

Exam Tip: Clearly write the subtraction step of the common middle segment BC from both sides to receive full reasoning marks.

5 Mark Questions

Question 39. Prove that an equilateral triangle can be constructed on any given line segment.
Answer: Let AB be a given line segment of specified length.
1. Using Euclid's Postulate 3 ("A circle can be drawn with any centre and any radius"), construct a circle with centre A and radius AB.
2. Similarly, draw a second circle with centre B and radius BA.
3. Let these two circles intersect each other at a point C.
4. Using Euclid's Postulate 1 ("A straight line may be drawn from any one point to any other point"), draw straight line segments joining A to C and B to C, forming triangle ABC.
5. In the circle centered at A, segments AB and AC are both radii:
\[ \text{AB} = \text{AC} \quad \text{--- (1)} \]
6. In the circle centered at B, segments AB and BC are both radii:
\[ \text{AB} = \text{BC} \quad \text{--- (2)} \]
7. Applying Euclid's Axiom 1: "Things which are equal to the same thing are equal to one another."
From (1) and (2), both AC and BC are equal to AB, which implies:
\[ \text{AC} = \text{BC} \]
Therefore:
\[ \text{AB} = \text{BC} = \text{AC} \]
Since all three sides of triangle ABC are equal in length, triangle ABC is an equilateral triangle constructed on segment AB.
Hence proved.
In simple words: Draw two circles centered at each end of the segment with radius equal to the segment. Where they cross gives the third corner, making all three sides equal.

Exam Tip: Be sure to explicitly cite Postulates 1 and 3 along with Axiom 1 during the proof to secure full marks.

Question 40. Does Euclid 's fifth postulate imply the existence of parallel lines? Explain. Any formulation the student gives should be discussed in the class for its validity.
Answer: Yes, Euclid's fifth postulate directly implies the existence of parallel lines.
Consider a transversal line \( l \) falling across two straight lines \( m \) and \( n \).
Let the interior angles on one side of line \( l \) be denoted as \( \angle 1 \) and \( \angle 2 \), and the interior angles on the opposite side as \( \angle 3 \) and \( \angle 4 \).
According to Euclid's fifth postulate, if the sum of interior angles on one side is strictly less than two right angles (\( \angle 1 + \angle 2 < 180^\circ \)), the lines will meet when extended on that side.
Now, suppose the sum of interior angles on one side equals exactly two right angles:
\[ \angle 1 + \angle 2 = 180^\circ \]
Since the adjacent angles form linear pairs along line \( l \):
\[ (\angle 1 + \angle 3) + (\angle 2 + \angle 4) = 180^\circ + 180^\circ = 360^\circ \]
\[ (\angle 1 + \angle 2) + (\angle 3 + \angle 4) = 360^\circ \]
Substituting \( \angle 1 + \angle 2 = 180^\circ \):
\[ 180^\circ + (\angle 3 + \angle 4) = 360^\circ \implies \angle 3 + \angle 4 = 180^\circ \]
Since the sum of interior angles is not less than \( 180^\circ \) on either side of the transversal, the lines cannot intersect when produced in either direction.
Lines that lie in the same plane and never intersect, regardless of how far they are extended, are by definition parallel lines.
Hence, Euclid's fifth postulate confirms the existence of parallel lines.
In simple words: When the interior angles on both sides add up to exactly 180 degrees, the lines never meet in either direction, which proves parallel lines exist.

Exam Tip: Show that both pairs of consecutive interior angles sum to 180° so that intersection is ruled out in both directions.

Question 41. State Euclid's five postulates.
Answer: Euclid's five geometric postulates are as follows:
1. Postulate 1: A straight line may be drawn from any one point to any other point.
2. Postulate 2: A terminated line can be produced indefinitely.
3. Postulate 3: A circle can be drawn with any centre and any radius.
4. Postulate 4: All right angles are equal to one another.
5. Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
In simple words: These five fundamental geometric rules describe drawing straight lines between points, extending segments forever, creating circles of any size, the equality of all right angles, and when two lines must cross.

Exam Tip: Write each postulate with its correct official number (1 to 5) and keep the wording as close to Euclid's original phrasing as possible.

Question 42. Which of the following statements are true and which are false? Give reasons for your answers.
(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, if AB = PQ and PQ = XY, then AB = XY.
Fig.
Answer:
(i) False. Infinitely many different lines can be drawn passing through any single point in a plane.
(ii) False. According to Euclid's axiom, given two distinct points, there exists one and only one unique line that passes through them.
(iii) True. By Euclid's Postulate 2, a terminated line (line segment) can be extended without limit in either direction to form a line.
(iv) True. Equal circles enclose identical surface areas; when placed over one another, their boundaries and centres coincide completely, which means their radii must be equal.
(v) True. According to Euclid's Axiom 1, "Things which are equal to the same thing are equal to one another." Since both AB and XY are equal to segment PQ, it follows that \( \text{AB} = \text{XY} \).
In simple words: Endless lines pass through one point, but only one line passes through two points. Line segments can extend forever, equal circles have equal radii, and two segments equal to the same segment are equal to each other.

Exam Tip: Always provide the specific axiom or postulate number as the justification when answering true/false questions in Euclid's geometry.