Important Questions Class 9 Maths Chapter 5
Important Questions Class 9 Mathematics Chapter 5 – Introduction to Euclid’s Geometry.
MathematicsChapter 5 of Class 9 introduces students to the topic of Introduction to Euclid’s Geometry. The term Geometry sees its origin in the Greek word ‘Geo’ means earth, and ‘Metrein’ means to measure. It is an ancient branch of Mathematics examined by different civilizations at various times. Euclid’s geometry is the study of solids and planes established on the axioms and postulates given by the Egyptian mathematician Euclid. It primarily deals with points, lines, circles, curves, angles, planes, solids, and so on.
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Important Questions Class 9 Mathematics Chapter 5 – With Solutions
Our in-house Mathematics faculty experts have collected an entire list of Important Questions Class 9 Mathematics Chapter 5 by referring to various sources. For every question, the faculty experts have prepared a detailed explanation that will enable students to understand the concepts used in each question. Also, the questions are picked in a way that would cover all the topics from the syllabus. So by working out all the questions from our question bank, students will be able to revise the chapter and comprehend their strong and weak points. And improvise by focusing on weak areas a little more to get the hang of the chapter.
Some of the questions and answers from our question bank of Mathematics Class 9 Chapter 5 Important Questions are given below:
Question 1: What are the five postulates of Euclid’s Geometry?
Answer 1: Euclid’s postulates are:
- A straight line may be drawn from one point to any other point.
- A required terminated line can be produced indefinitely.
- A required circle can be drawn with any centre and radius.
- All right angles are equal.
- Suppose a given straight line intersects on two straight lines making the interior angles on the exact side of it brought together less than two right angles. In that case, the two straight lines, if produced indefinitely, meet on that side on which the sum of the given angles is less than two right angles.
Question 2: Which of the given following assertions are true and which are false? Give explanations for your solutions.
(i) Just one line can pass through a given single point.
(ii) An infinite number of lines pass through two required distinct points.
(iii) A given terminated line can be created indefinitely on both sides.
(iv) If two circles are equal, their radii are equal.
(v) In the figure alongside , if AB = PQ and PQ = XY, then AB = XY.
Answer 2:
(i) The above statement is False
Reason: If we draw a point O on the surface of a paper. Utilising pencil and scale, we can draw an infinite number of straight lines passing
through O.
(ii) The above statement is False
Reason: In the following figure, multiple straight lines pass through P. There are numerous lines, passing through Q. But there is one line which is passing through P and Q.
(iii)The above statement is True
Reason: Postulate 2 says, “A terminated line can be produced indefinitely.”
(iv)The above statement is True
Reason: Postulate 3 which mentions a circle can be drawn with any centre and any radius.
Thus, their radii will coincide or be equivalent.
(v) The above statement is True
Reason : According to Euclid’s axiom, things which are equal to the same thing are equal to one another.
Question 3: Describe parallel lines. Are there other representations that need to be specified first? What are they, and how might you express them?
Answer 3:The required two coplanar lines in a plane not intersecting are also said to be parallel lines. The other term intersecting, is undefined.
Question 4: Define the following terms individually. Are there other terms that ought to be specified first? What are they, and how will you describe them?
(i)The parallel lines
(ii) The perpendicular lines
(iii) The line segment
(iv) The radius of a circle
(v) The square
Answer 4: Yes, we need to know the words like point, line, ray, angle, plane, circle and quadrilateral, etc., before describing the necessary words.
Descriptions of the necessary words are given below:
(i) The parallel lines:
The two lines l and m in a plane are expressed as parallel if they have no common point, and we note them as l ॥ m.
The distance between the lines always remains the same.
Lines l and m are parallel on a plane surface.
(ii) The perpendicular lines:
The two lines, p and q, lying in the same plane, are expressed to be perpendicular if they intersect each other at right angle, and we note them as p ⊥ q.
Lines q and p are perpendicular to each other.
(iii)The line Segment:
A required line segment is a part of a line and has a definite length. It has two endpoints. In the figure, a line segment has endpoints A and B. It cannot be extended. It is written as
AB or BA
(iv) The radius of a circle :
The length from the centre to a point on the circle is called the radius of the circle. Radius helps in calculation of diameter, circumference and area.In the figure alongside, P is the centre, Q is a point on the circle, and PQ is the radius.
(v) Square :
A quadrilateral in which all the four angles are right angles and all the four sides are equal is called a square. Opposite sides of the square are parallel. Two diagonals bisect each other at a right angle. Given the figure, PQRS is a square.
Question 5: Define perpendicular lines. Are there other words that need to be described first? What are they, and how may you define them?
Answer 5:The two coplanar (in a plane) lines are perpendicular if the required angle between them at the given point of intersection is one right angle. In other words, the point of intersection and one right angle are undefined.
Question 6: In figure alongside, if AC = BD, then prove that AB = CD.
Answer 6: Given: AC = BD
⇒ AB + BC = BC + CD
Subtracting BC from both sides, we get
AB + BC – BC = BC + CD – BC
[When the equivalents are subtracted from equals, remainders are equal]
⇒ AB = CD
Question 7: Define the line segment. Are there other terms that ought to be defined first? What are they, and how will you define them?
Answer 7:A line segment PQ of a line ‘l is the continuous part of line I with endpoints P and Q.
Here, the continuous part of the line ‘l is undefined.
Question 8: Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.
Answer 8: Yes. If a required straight line l falls on two lines m and n such that the sum of the given interior angles on one side of l is two right angles, then by Euclid’s fifth postulate, the lines m and n will not meet on this side of l. Furthermore, we understand that the sum of the given interior angles on the other side of the line l will be the two given right angles too. Hence, they will not meet on the other side also.
∴ The lines m and n never meet; they are parallel. In figure, if AC = BD, then verify that AB = CD.
Question 9: Solve the required equation a – 15 = 25 and state which axiom you use here.
Answer 9: a – 15 = 25
By adding 15 to both sides, we obtain
a – 15 + 15 = 25 + 15 [using Euclid’s second axiom]
a = 40
Question 10: Assume the two ‘postulates’ given below
(i) Given any two distinct points, A and B, there exists a third point, C, between A and B.
(ii) At least three points are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow Euclid’s postulates? Explain.
Answer 10: Yes, these postulates have undefined terms such as ‘Point and Line. Furthermore, these postulates are consistent because they deal with two different situations as
(i) states that required two points A and B, there is a point C on the line between them. Whereas
(ii) states that, given points A and B, you can take point C, not lying on the line through points A and B.
No, these postulates do not obey Euclid’s postulates. Nevertheless, they abide by the axiom, “Given two distinct points, there is a unique line that passes through them.”
Question 11: Ram and Ravi have the exact weight. If they ever gain weight by 2 kg, how will their new weights be compared?
Answer 11: Let x kg be the weight of each of Ram and Ravi.
On adding 2 kg,
The weight of Ram and Ravi will be (x + 2) kg each.
According to Euclid’s second axiom, the wholes are equal when equals are added to equals.
So, the weight of Ram and Ravi are equal.
Question 12: In question 4, point C is called a mid-point line segment AB. Verify that every line segment has one and only one mid-point.
Answer 12: Let the given line AB have two midpoints,’ C’ and ‘D’.
Image source: https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/grade-09-ch05-13-1564724464.png
AC = 1/2AB ……(i)
and AD = 1/2AB ……(ii)
Subtracting (i) from (ii), we have
AD – AC = 1/2AB−1/2AB
or AD – AC = 0 or CD = 0
∴ C and D coincide.
Therefore, each line segment has one and only one mid-point.
Question 13: If point C be the mid-point of a line segment AB, then write the relation among AC, BC and AB.
Answer 13: Here, C is the mid-point of AB
AC = BC
AC = BC = 1/2AB
Question 14: Read the required statements:
A given equilateral triangle is a polygon made up of three line segments, of which two line segments are equivalent to the third one, and all its angles are 60° each.
Explain the words used in this description that you feel are essential. Are there any indefinite words in this? Can you explain that all sides and all angles are equivalent in an equilateral triangle?
Answer 14: The terms that need to be defined are
(i) The given polygon A closed figure bounded by three or more line segments.
(ii) The given line segment is part of a line with two endpoints.
(iii) The given line Undefined term.
(iv) The given point Undefined term.
(v) Angle =A given figure formed by two rays with one common initial point.
(vi) Acute angle =Angle whose measure is from 0° to 90°.
Here undefined terms are line and point.
All the angles of any given equilateral triangle are 60° each (given).
The two line segments are equal to the third one (given).
Hence, all three sides of an equilateral triangle are equal because, according to Euclid’s axiom, things equal to the same thing are equal.
Question 15: If point P is MN’s mid-point and C is MP’s mid-point, then write the relation between MC and MN.
Answer 15: Here, P is MN’s mid-point, and C is MP’s mid-point.
∴ MC = 1/4 MN
Question 16: Review the required statements
“The two intersecting lines cannot be perpendicular to the same line.” Inspect whether it is an equivalent version to Euclid’s fifth postulate.
Image source: https://d1hj4to4g9ba46.cloudfront.net/questions/270844_242473_ans_017df27fc127473eaa453573f74433ef.png
Answer 16: Two equivalent versions of Euclid’s fifth postulate are
(i) For every line l and each point P not lying on Z, there exists a special line m passing through P and parallel to Z.
(ii) The two distinct intersecting lines cannot be parallel to the same line.
The above two statements show that the given statement is not an equivalent version of Euclid’s fifth postulate.
Question 17:How many lines pass through two distinct points?
Answer 17: One and only one line passes through two distinct points.
Question 18: Read the required statements, which are taken as axioms
(i) If a transversal intersects the two required parallel lines, then corresponding angles are not necessarily equal.
(ii) If a transversal intersects the two required parallel lines, alternate interior angles are equivalent.
Is this system of axioms consistent? Explain your answer.
Image source: https://d1hj4to4g9ba46.cloudfront.net/questions/146722_77779_ans.png
Answer 18: A system of the axiom is said to be constant if there is no statement deduced from these axioms such that it contradicts any axiom. We understand that if a given transversal intersects the two parallel lines, every pair of corresponding angles is equal, which is a theorem. So, Statement I is incorrect and not an axiom.
Furthermore, if a transversal intersects the two given parallel lines, every alternate interior angle pair is identical. It is also a theorem. So, Statement II is true and an axiom. . Therefore, in given statements, the first is false, and the second is an axiom.
Therefore, the required system of axioms is not constant.
Question 19: In the figure, if AB = CD, then prove that AC = BD. Also, write Euclid’s axiom used to prove it.
Answer 19:
Image source:https://d1hj4to4g9ba46.cloudfront.net/questions/463630.jpg
AC=BD … (i)
AC=AB+BC ……(ii) (The point B lies between A and C)
BD=BC+CD … (iii) (The point C lies between B and D)
Now,
replacing (ii) and (iii) in (i), we get
AB+BC=BC+CD
AB+BC–BC=CD
AB=CD
Therefore, AB=CD.
Question 20: Read the following two statements, which are taken as axioms:
(i) If the two lines intersect, then the required vertically opposite angles are not equal.
(ii) If a given ray stands on a line, then the sum of two adjacent angles, so formed, is equal to 180°.
Is the given system of axioms consistent? Explain your answer.
Answer 20:
We understand that if two lines intersect, the vertically opposite angles are equal. It is a theorem. The given Statement, (i), is false and not an axiom.
Furthermore, if a ray stands on a line, then the sum of two adjacent angles is equal to 180°. It is an axiom. The given Statement (ii) is true and has an axiom.
Therefore, the first is false in given statements, and the second is an axiom. Accordingly, the given system of axioms is not consistent.
Question 21: In the given figure, name the following :
(i) Four collinear points
(ii) Five rays
(iii) Five line segments
(iv) Two pairs of non-intersecting line segments.
Answer 21: (i) The four collinear points are D, E, F, G and H, I, J, K
(ii) The five rays are DG, EG, FG, HK, and IK.
(iii) The five line segments are DH, EI, FJ, DG, and HK.
(iv) The two pairs of non-intersecting line segments are (DH, EI) and (DG, HK).
Question 22: Read the following axioms
(i) The things equivalent to the identical thing are equal.
(ii) The wholes are equivalent if equals are added to equals.
(iii) Things which are double of the same things are equal.
Inspect whether the given system of axioms is consistent or inconsistent.
Thinking Process
To inspect whether the required system is consistent or inconsistent, we have to find whether we can deduce a statement from these axioms that contradicts any axiom.
Answer 22: Some of Euclid’s axioms are
(i) The things equal to the same thing are equal.
(ii) The wholes are equal if equals are added to equals.
(iii) The things which are double the exact things are equal.
Therefore, the given three axioms are Euclid’s axioms. Here we cannot conclude any statement from these axioms which contradicts any axiom. The given system of axioms is consistent.
Question 23: In the given figure, AC = DC and CB = CE. Show that AB = DE. Write Euclid’s axiom to support this.
Answer 23: We have
AC = DC
CB = CE
Utilising Euclid’s axiom 2, if equals are added to equals, then the wholes are equal.
⇒ AC + CB = DC + CE
⇒ AB = DE.
Question 24: In the figure, it is given that AD=BC. By which Euclid’s axiom can it be verified that AC = BD?
Answer 24: We can verify it by Euclid’s axiom 3. “If equivalents are subtracted from equals, the remainders are equal.”
AD = BC
AD – CD = BC – CD
AC = BD
Question 25: In the adjoining figure, if OX = 1/2 XY, PX = 1/2 XZ and OX = PX, show that XY = XZ.
Answer 25: Given OX= 1/2 XY
2 OX = XY ……….( equation i )
PX= 1/2 XZ
2 PX = XZ …….( equation ii )
and OX = PX …( equation iii )
As per Euclid’s axiom, things which are double the exact things are equal.
On multiplying ( equation iii) by 2, we acquire
2 OX = 2 PX
XY=XZ. [from ( equation i ) and ( equation ii )]
Question 26: If the required three points A, B and C are on a given line and B lies between A and C (see figure alongside), then verify that AB + BC = AC.
Answer 26: In the given figure, AC coincides with AB + BC. Also, Euclid’s axiom 4 states that things
that coincide are equal. So, it is evident that:
AB + BC = AC.
Question 27: In the given figure, AB = BC, BX = BY, show that
AX = CY.
Answer 27: Given: AB = BC
and BX = BY
By utilising Euclid’s axiom 3, equals subtracted from equals, then the remainders are equal. We acquire
AB – BX = BC – BY
AX = CY
Question 28: In the figure alongside, if AB = PQ, PQ = XY, then AB = XY. Express whether it is True or False. Explain your answer.
Answer 28: True. ∵ Euclid’s first axiom states, “Things which are equivalent to the same thing are also equal to one another”.
∴ AB = PQ and XY = PQ
Therefore, AB = XY
Question 29: We have AC = DC and CB = CE in the adjoining figure. Show that AB = DE.
Answer 29: Given, AC = DC …(i)
and CB = CE …(ii)
As per Euclid’s axiom, if equals are added to equals, then the wholes are also equal.
Add Eqs. (i) and (ii), we obtain
AC + CB = DC + CE
=> AB = DE
Question 30: In the given figure, if ∠1 = ∠3, ∠2 = ∠4 and ∠3 = ∠4, write the relation between ∠1 and ∠2 using Euclid’s axiom.
Answer 30: Here, ∠3 = ∠4, ∠1 = ∠3 and ∠2 = ∠4. Euclid’s first axiom says that the things which are equal to the same things are equal to one another. . So ∠1 = ∠2.
Question 31: The adjoining figure has ∠ABC = ∠ACB and ∠3 = ∠4. Show that BD = DC.
Answer 31: Given, ∠ABC = ∠ACB …(i)
and ∠4 = ∠3 …(ii)
As per Euclid’s axiom, if equivalents are subtracted from equivalents, then remainders are also equivalent.
On deducting Eq. (ii) from Eq. (i), we get
∠ABC – ∠4 = ∠ACB – ∠3 =>∠1 = ∠2
Currently, in ABDC, ∠1=∠2
=> DC =BD [the required sides opposite to equal angles are equal]
BD = DC.
Question 32: In the given figure, we have ∠1 = ∠2, ∠3 = ∠4. Show that ∠ABC = ∠DBC. State Euclid’s Axiom is used.
Answer 32: Here, 1 = ∠2 and ∠3 = ∠4.
By utilising Euclid’s Axiom 2.
If equivalents are added to
equivalents, then the wholes are equivalent.
∠1 + ∠3 = ∠2 + ∠4
∠ABC = ∠DBC.
Question 33: In the figure alongside, we have BX and 1/2 AB =1/2 BC. Show that BX = BY.
Image source: https://d1hj4to4g9ba46.cloudfront.net/questions/1794461_5291335a02d3469fbc25f896029da5ff.png
Answer 33: Here, BX = 1/2 AB and BY = 1/2 BC …( Equation i ) [given]
Furthermore, AB = BC [given]
⇒ 1/2AB = 1/2BC …( equation ii )
[∵ Euclid’s seventh axiom states that things which are halves of the same thing are equivalent to one another]
From the given ( equation i ) and ( equation ii ), we have BX = BY.
Question 34: In the figure alongside, AC = XD, C is the mid-point of AB and D is the mid-point of XY. Using Euclid’s axiom, show that AB = XY.
Image source: http://images.topperlearning.com/topper/questions/212239_image067.jpg
Answer 34: ∵ C is the mid-point of AB
AB = 2AC
Also, D is the mid-point of XY
XY = 2XD
Euclid’s sixth axiom states, “Things which are double of same things are equal to one another.”
∴ AC = XD = 2AC = 2XD
Therefore, AB = XY
Question 35: For presented four distinct points in a given plane, find out the number of lines that can be drawn through :
(i) All four points are collinear.
(ii) Three of the four points are collinear.
(iii) No three of the four points are collinear.
Answer 35:
(i) Assume the points shown are A, B, C and D.
When all the four points are collinear :
One line AD.
(ii) When three of the four given points are collinear :
Four lines
Here, we have four lines AB, BC, BD and AD (four).
(iii) When no three of the four given points are collinear :
Six lines Here, we have
AB, BC, AC, AD, BD and CD (six).
Question 36: Express that : length AH > sum of lengths of AB + BC + CD.
Answer 36: We acquire
AH = AB + BC + CD + DE + EF + FG + GH
AB + BC + CD is a component of AH.
As per Euclid’s axiom, “The whole is greater than the part”
⇒ AH > AB + BC + CD
Therefore, the length AH > sum of lengths AB + BC + CD.
Question 37: The three given lighthouse towers are found at points A, B and C in the section of a national forest to preserve animals from hunters in the forest department, as shown in the figure. Which value is the department displaying by locating extra towers? How many straight lines may be drawn from A to C? Note the Euclid Axiom, which states the required result. Provide one more. Postulate.
Answer 37:
(i) One certain line can be drawn from A to C. Euclid’s Postulate states, “A straight line can be drawn from any required point to any other point:”
(ii)Another postulate states, “A circle may be represented with any centre and radius.” Wildlife is a component of our environment, and the preservation of every element is essential for ecological balance.
Question 38: In the adjoining figure, we have ∠1 =∠3 and ∠2 = ∠4. Show that ∠A = ∠C.
Answer 38:
Given,∠1 = ∠3 …(i)
and ∠2 = ∠4 …(ii)
According to Euclid’s axiom 2, if equals are added to equals, then wholes are also equal.
On adding Eqs. (i) and (ii), we get
∠1 + ∠2 = ∠3 +∠4
=> ∠A = ∠C
Question 39: Two salesmen make equal sales during the month of August. In September, each salesman doubles his sales for the month in August. Compare their sales in September. Solve using Euclid’s axiom.
Answer 39:
Let the equal sale of two salesmen in August be x.
In September, each salesman doubles his sales for the month of August.
Thus, the sale of the first salesman is 2x and the sale of the second salesman is 2x.
According to Euclid’s axioms, things which are double of the same things are equal to one another.
So, in September their sales are again equal.
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Revision by practising and solving questions is the key to scoring good marks in Mathematics. The Mathematics of grades 8, 9, and 10 forms the pillar for future Class 11 and 12 and many other education streams. We suggest students solve questions from our question bank of Important Questions Class 9 Mathematics Chapter 5. Our team has collated exam-oriented questions and prepared their step-by-step solutions. By properly solving questions and going through the given answers, students will get the confidence to solve any other complicated questions from the chapter introduction to Euclid’s Geometry.
Here are a few benefits of solving questions regularly from our question bank of Important Questions Class 9 Mathematics Chapter 5:
- By referring to the detailed step-by-step explanations given in our solutions, students will better learn about all the concepts, theorems and formulas covered in Chapter 5 of Class 9 Mathematics syllabus.
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Q.1 Mention Five postulates of Eulid.
Marks:4
Ans
Postulate 1 : A straight line may be drawn from any one point to any other point.
Postulate 2 : A terminated line can be produced indefinitely.
Postulate 3 : A circle can be drawn with any centre and any radius.
Postulate 4 : All right angles are equal to one another.
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.
Q.2 In Fig., if AC = BD then prove that AB = CD.
Marks:3
Ans
AC = AB + BC …(2) (Point B lies between A and C)
BD = BC + CD …(3) (Point C lies between B and D) (3)
Since AC = BD
AB + BC = BC + CD
So, AB = CD (Subtracting equals from equals)
Q.3 Prove that an equilateral triangle can be constructed on any given line segment.
Marks:4
Ans
According to this statement, a line segment of any length is given, say AB
Now, draw a circle with centre A and radius AB. Similarly draw another circle with centre B and radius BA. These two circles meet at C. Now join AC and BC to form triangle ABC.
Now, AB=BC [Radii of the same circle]
AB=AC [Radii of the same circle]
With the help of Euclid’s axiom that things which are equal to the same thing are equal to one another, so we can conclude that AB=BC=CA.
So triangle ABC is an equilateral triangle.
Q.4 State two equivalent versions of Ecluid’s fifth postulate.
Marks:2
Ans
(i) For every line l and for every point P not lying on l, there exists a unique line n passing through P and parallel to l.
(ii) Two distinct intersecting lines cannot be parallel to the same line.
Q.5 If A, B and C are three points on a line, and B lies between A and C then prove that AB + BC = AC.
Marks:2
Ans
In above figure, AC coincides with AB + BC.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another.
So, AB + BC = AC
