CBSE Class 9 Mathematics Introduction to Euclids Geometry Assignment Set 03

Chapter - 5

(Introduction to Euclid's Geometry)

The Greeks developed geometry in a systematic manner. Euclid (300 B.C.), a Greek mathematician and father of geometry, introduced the method of proving mathematical results by using deductive logical reasoning and previously proved results. The geometry of plane figures is known as Euclidean Geometry.

Axioms:

The basic facts which are taken for granted without proof are called axioms. Some of Euclid's axioms are:

(i) Things which are equal to the same thing are equal to one another, i.e., \(a = b\) and \(b = c \implies a = c\)
(ii) If equals are added to equals, the wholes are equal, i.e., \(a = b \implies a + c = b + c\)
(iii) If equals are subtracted from equals, the remainders are equal, i.e., \(a = b \implies a - c = b - c\)
(iv) Things which coincide with one another are equal to one another.
(v) The whole is greater than the part.

Postulates:

Axioms are general statements, while postulates are axioms relating to a particular field. Euclid'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.
e.g., (i) \(4 + 9 = 6\) is a false sentence, so it is a statement.
(ii) Sanjay 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.
e.g., (i) The sum of the angles of a triangle is \(180^\circ\).
(ii) The angles opposite to equal sides of a triangle are equal.

Corollary: A result deduced from a theorem is called its corollary.

Section - A

Question 1. Euclid stated that if equals are subtracted from equals, the remainders are equals in the form of
(a) an axiom
(b) a postulate
(c) a definition
(d) a proof
Answer: (a) an axiom
Euclid's third axiom states that if equals are subtracted from equals, the remainders are equal.
In simple words: Taking equal amounts away from equal amounts leaves equal amounts, which is a basic axiom.

Exam Tip: Remember that axioms are general assumptions across all mathematics, not restricted to geometry.

Question 2. 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) a postulate
This statement is Euclid's fourth postulate, which deals specifically with geometrical figures (right angles).
In simple words: The fact that all right angles measure \(90^\circ\) and are equal is Euclid's fourth postulate.

Exam Tip: Distinguish carefully between Euclid's axioms (general mathematics) and postulates (specific geometry rules).

Question 3. Which of the following needs a proof:
(a) Theorem
(b) Axiom
(c) Definition
(d) Postulate
Answer: (a) Theorem
A theorem is a mathematical statement that requires logical proof using axioms, postulates, and previously proved theorems.
In simple words: Theorems must be proven, whereas axioms and postulates are accepted without proof.

Exam Tip: Axioms and postulates are starting assumptions, while theorems are conclusions derived from them.

Question 4. The number of dimensions a solid has:
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (c) 3
A solid has three dimensions: length, breadth, and height/thickness.
In simple words: Physical 3D objects have length, width, and height (3 dimensions).

Exam Tip: Solid = 3 dimensions, Surface = 2 dimensions, Line = 1 dimension, Point = 0 dimensions.

Question 5. The number of dimensions a surface has:
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (b) 2
A surface has two dimensions: length and breadth.
In simple words: Surfaces (like a flat sheet of paper) have only length and width (2 dimensions).

Question 6. The number of dimensions a point has:
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0
A point is defined as that which has no part, meaning it has zero dimensions.
In simple words: A dot on paper has position but no length, width, or height, so it has 0 dimensions.

Exam Tip: A point only indicates position.

Question 7. Which one of the following statements is true?
(a) Only one line can pass through a single point.
(b) There are infinite numbers of lines which pass through two distinct points.
(c) Two distinct lines cannot have more than one point in common.
(d) If two circles are equal, then their radii are not equal.
Answer: (c) Two distinct lines cannot have more than one point in common.
By Euclid's axiom, two distinct intersecting lines cannot intersect at more than one point. Option (a) is false because infinite lines pass through one point; option (b) is false because only one line passes through two distinct points; option (d) is false because equal circles have equal radii.
In simple words: Two straight lines can cross each other at most at one single point.

Exam Tip: Memorize Euclid's basic postulates regarding lines and points.

Question 8. Euclid divided his famous treatise "The Elements" into
(a) 13 chapters
(b) 12 chapters
(c) 11 chapters
(d) 9 chapters
Answer: (a) 13 chapters
Euclid's monumental work "The Elements" was divided into 13 books or chapters.
In simple words: Euclid wrote his famous book in 13 parts (chapters).

Exam Tip: This is a standard historical fact often asked in exams.

Question 9. Thales belongs to the country:
(a) Babylonia
(b) Egypt
(c) Greece
(d) Rome
Answer: (c) Greece
Thales was a famous Greek mathematician and philosopher.
In simple words: Thales was from Greece.

Exam Tip: Associate Thales and Euclid with ancient Greece.

Section - B

Question 10. If \(AB = CD\), can you say that \(AC = BD\)? Give reasons for your answer.
Answer: Yes, \(AC = BD\).
Given: \(AB = CD\)
Adding \(BC\) to both sides (using Euclid's axiom: if equals are added to equals, the wholes are equal):
\(AB + BC = CD + BC\)
From the figure, \(AB + BC = AC\) and \(CD + BC = BD\).
Therefore, \(AC = BD\).
In simple words: Adding the same middle segment \(BC\) to two equal segments \(AB\) and \(CD\) results in equal total lengths \(AC\) and \(BD\).

Exam Tip: Clearly state which Euclid axiom you are using in geometry proof steps.

Question 11. In how many points can two distinct lines intersect?
Answer: Two distinct lines can intersect at a maximum of one point.
In simple words: Two straight lines can cross each other at only one spot.

Exam Tip: Distinct intersecting lines share exactly one common point.

Question 12. In how many lines can two distinct planes intersect?
Answer: Two distinct planes intersect in a single line.
In simple words: When two flat walls or planes meet, they form a straight line of intersection.

Exam Tip: Think of the intersection of two adjacent walls in a room, which meets along a straight vertical line.

Section - C

Question 13. If a point \(C\) lies between two points \(A\) and \(B\) such that \(AC = CB\), then prove that \(AC = \frac{1}{2}AB\). Explain by drawing the figure.
Answer:
Given: Point \(C\) lies between \(A\) and \(B\) such that \(AC = CB\).
From the line segment, we have:
\(AB = AC + CB\)
Since \(CB = AC\), we can substitute \(AC\) for \(CB\):
\(AB = AC + AC\)
\(AB = 2AC\)
Dividing both sides by 2 gives:
\(AC = \frac{1}{2}AB\)
In simple words: If a line segment is divided into two equal halves, each half is half the total length.

Exam Tip: Always draw a neat line segment diagram showing points \(A\), \(C\), and \(B\) in order.

Question 14. In the figure, if \(AC = BD\), prove that \(AB = CD\).
Answer:
Given: \(AC = BD\)
From the given line segment arrangement, we can write:
\(AC = AB + BC\)
\(BD = BC + CD\)
Substitute these into the given equation \(AC = BD\):
\(AB + BC = BC + CD\)
Subtracting \(BC\) from both sides (using Euclid's axiom: if equals are subtracted from equals, remainders are equal):
\(AB = CD\)
In simple words: Subtracting the common overlapping part \(BC\) from two equal lengths \(AC\) and \(BD\) leaves the remaining outer parts \(AB\) and \(CD\) equal.

Exam Tip: Mention the subtraction axiom clearly when eliminating common lengths.

Question 15. If \(C\) is called a midpoint of line segment \(AB\), prove that every line segment has one and only one midpoint.
Answer:
Let \(AB\) be a given line segment. Let us assume, if possible, that there are two midpoints of \(AB\), say \(C\) and \(D\).
If \(C\) is a midpoint of \(AB\), then:
\(AC = CB\) and \(AC = \frac{1}{2}AB\) ---(1)
If \(D\) is also a midpoint of \(AB\), then:
\(AD = DB\) and \(AD = \frac{1}{2}AB\) ---(2)
From (1) and (2), we get:
\(AC = AD\)
This means points \(C\) and \(D\) must coincide at the exact same position.
Therefore, every line segment has one and only one midpoint. Proved.
In simple words: By assuming two midpoints and showing they must occupy the exact same point, we prove a line segment can only have one midpoint.

Exam Tip: Use the method of proof by contradiction (assuming two distinct midpoints and showing a logical collision).

Question 16. Define the following terms: (i) Parallel lines (ii) Concurrent lines (iii) Intersecting lines
Answer:
(i) Parallel lines: Two lines in a plane that do not intersect anywhere and remain at a constant perpendicular distance apart from each other are called parallel lines.
(ii) Concurrent lines: Three or more lines passing through the same single point are called concurrent lines.
(iii) Intersecting lines: Two lines that cross each other at a common point are called intersecting lines.
In simple words: Parallel lines never meet, concurrent lines all cross through one common point, and intersecting lines cross each other.

Exam Tip: Learn exact standard definitions as they are frequently tested in theory questions.

Question 17. State Euclid's any three postulates.
Answer:
1. A straight line may be drawn from any one point to any other point.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any centre and any radius.
In simple words: Any two points can be joined by a straight line, a line segment can be extended endlessly in both directions, and a circle can be drawn using any point as center and any opening as radius.

Exam Tip: Memorize all 5 postulates verbatim as they form the foundation of this chapter.

Question 18. State Euclid's any three axioms.
Answer:
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. The whole is greater than the part.
In simple words: Quantities equal to a common quantity are equal to each other; adding equal amounts gives equal results; and any complete whole is always bigger than any of its individual parts.

Exam Tip: Be careful not to confuse Euclid's axioms with postulates when writing answers.