NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles
NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles is curated and compiled by the expert team of professionals at GFG to assist students in resolving questions related to lines and angles they may have as they go through problems from the NCERT textbook.
This chapter Lines and Angles mainly covers the characteristics of the angles created when two or more lines cross one another. Additionally, it gives a fundamental comprehension of the definitions of various terminology used in geometry. The many kinds of angles, pairs of angles, transversals, parallel lines, the angle sum property of triangles, and axioms related to these ideas are all taught to students.
Class 9 Maths NCERT Solutions Chapter 6 Exercises: |
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NCERT Maths Solutions Class 9 Exercise 6.1 â€“ 6 Questions & Solutions (5 Short Answers, 1 Long Answer) |
NCERT Maths Solutions Class 9 Exercise 6.2 â€“ 3 Questions & Solutions (3 Short Answers, 3 Long Answers) |
NCERT Maths Solutions Class 9 Exercise 6.3 â€“ 6 Questions & Solutions (5 Short Answers, 1 Long Answer) |
Lines and Angles – Exercise 6.1
Question 1. In the given figure, lines AB and CD intersect at O. If âˆ AOC + âˆ BOE = 70Â° and âˆ BOD = 40Â°, find âˆ BOE and reflex âˆ COE?
Solution:
Given, AB and CD are straight lines.
âˆ AOC + âˆ BOE = 70Â° —-eq(i)
âˆ BOD = 40Â° —-eq(ii)
Since, AB is a straight line, the sum of all angles made on it is 180Â°
â‡’ âˆ AOC + âˆ COE + âˆ BOE = 180Â° —eq(iii)
We can rearrange this equation as,
â‡’ âˆ AOC + âˆ BOE + âˆ COE = 180Â°
â‡’ 70Â° + âˆ COE = 180Â° —from eq(i)
â‡’ âˆ COE = 180Â° – 70Â° = 110Â°
â‡’ âˆ COE = 110Â° —eq(iv)
Reflex âˆ COE = 360Â° – âˆ COE = 360Â° – 110Â° = 250Â°
Now, it is also given that CD is also a straight line, so the sum of all angles made on it is 180Â°
â‡’ âˆ COE + âˆ BOE + âˆ BOD = 180Â° —eq(v)
We can rearrange this equation as,
â‡’ âˆ COE + âˆ BOD + âˆ BOE = 180Â°
â‡’ 110Â° + 40Â° + âˆ BOE = 180Â° —from eq(ii) and eq(iv)
â‡’ 150Â° + âˆ BOE = 180Â°
â‡’ âˆ BOE = 180Â° – 150Â° = 30Â°
â‡’ âˆ BOE = 30Â°
Question 2. In the given figure, lines XY and MN intersect at O. Ifâˆ POY = 90Â° and a : b = 2 : 3, find c?
Solution:
Given, XY and MN are straight lines.
âˆ POY = 90Â° –eq(i)
a : b = 2 : 3 –eq(ii)
âˆ POM = a
âˆ XOM = b
âˆ XON = c
Taking XY as a straight line, so the sum of all angles made on it is 180Â°
â‡’ âˆ XOM + âˆ POM + âˆ POY = 180Â° —eq(iii)
â‡’ b + a + 90Â° = 180Â°
â‡’ 3x + 2x + 90Â° = 180Â° from eq(i) and eq(ii)
â‡’ 5x + 90Â° = 180Â°
â‡’ 5x = 180Â° – 90Â° = 90Â°
â‡’ 5x = 90Â°
â‡’ x = 18Â°
a : b = 2x : 3x = 2×18 : 3×18
a = 36Â°
b = 54Â°
Taking MN as a straight line so,the sum of all the angles made on it is 180Â°
â‡’ âˆ XOM + âˆ XON = 180Â°
â‡’ 54Â° + âˆ XON = 180Â° from above finding value
â‡’ âˆ XON = 126Â° or c = 126Â°
Question 3. In the given figure, âˆ PQR = âˆ PRQ, then prove that âˆ PQS = âˆ PRT?
Solution:
Given, âˆ PQR = âˆ PRQ
Taking ST is a straight line, so the sum of all angles made on it is 180Â°
â‡’ âˆ PQS + âˆ PQR = 180Â° —-eq(i)
also, âˆ PRQ + âˆ PRT = 180Â° —eq(ii)
By equating both the equations because RHS of both the equation is equal So, LHS will also be equal.
â‡’ âˆ PQS + âˆ PQR = âˆ PRQ + âˆ PRT
â‡’ âˆ PQS + âˆ PQR = âˆ PQR + âˆ PRT –[ Given in question âˆ PQR = âˆ PRQ ]
â‡’ âˆ PQS = âˆ PRT
Question 4. In the given figure, if x + y = w + z, then prove that AOB is a line?
Solution:
Given, x + y = w + z –eq(i)
We know that , sum of all angles made along a point is 360Â°
So, Taking O as a point âˆ AOC + âˆ BOC + âˆ BOD + âˆ AOD = 360Â°
â‡’ y + x + w + z = 360Â° from the given figure
â‡’ (x + y) + (x + y) = 360Â° from eq(i)
â‡’ 2x + 2y = 360Â°
â‡’ 2(x + y) = 360Â°
â‡’ x + y=180Â°
From this statement it is proved that AOB is a straight line because the sum of angles made on the line is 180Â°. So, AOB is a straight line.
Question 5. In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove thatâˆ ROS = (1/2) (âˆ QOS â€“ âˆ POS)?
Solution:
Given POQ is a straight line
So, the sum of all angles made on it is 180Â°
â‡’ âˆ POS + âˆ ROS + âˆ ROQ = 180Â°
â‡’ âˆ POS + âˆ ROS + 90Â° = 180Â° [given âˆ ROQ = 90Â°]
â‡’ âˆ POS + âˆ ROS = 90Â°
â‡’ âˆ ROS = 90Â° – âˆ POS –eq(i)
Now, âˆ ROS + âˆ ROQ = âˆ QOS [from figure]
â‡’ âˆ ROS + 90Â° = âˆ QOS
â‡’ âˆ ROS = âˆ QOS – 90Â° –eq(ii)
Now Adding both the equations eq(i) + eq(ii)
â‡’ âˆ ROS + âˆ ROS = 90Â° – âˆ POS + âˆ QOS – 90Â°
â‡’ 2âˆ ROS =(âˆ QOS – âˆ POS)
â‡’ âˆ ROS = (1/2) (âˆ QOS – âˆ POS)
Hence Verified!!!
Question 6. It is given that âˆ XYZ = 64Â° and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects âˆ ZYP, find âˆ XYQ and reflex âˆ QYP?
Solution:
From the drawn figure, it is clearly shown that XYP is a straight line.
So, âˆ XYZ + âˆ ZYQ + âˆ QYP = 180Â°
â‡’ 64Â°+ âˆ ZYQ + âˆ QYP = 180Â° [ given âˆ XYZ = 64Â°]
â‡’ 64Â° + 2âˆ QYP = 180Â° [ YQ bisect âˆ ZYP so, âˆ QYP = âˆ ZYQ]
â‡’ 2âˆ QYP = 180Â° – 64Â° = 116Â°
â‡’ âˆ QYP = 58Â°
So, Reflex âˆ QYP = 360Â° – 58Â° = 302Â°
Since âˆ XYQ = âˆ XYZ + âˆ ZYQ
â‡’ âˆ XYQ = 64Â° + âˆ QYP [ given âˆ XYZ = 64Â° and âˆ ZYQ = âˆ QYP]
â‡’ âˆ XYQ =64Â° + 58Â° = 122Â°
Thus, âˆ XYQ = 122Â° and Reflex âˆ QYP = 302Â°
Lines and Angles – Exercise 6.2
Question 1. In given figure, find the values of x and y and then show that AB || CD.
Solution:
After given names to the remaining vertices we get,
Now, Given âˆ AEP = 50Â°, âˆ CFQ = 130Â°
â‡’ âˆ EFD = âˆ CFQ [vertically opposite angles are equal]
â‡’ y = 130Â° [Given âˆ CFQ = 130Â°]
â‡’ y = 130Â° —eq(i)
Now, PQ is taking as straight line so, sum of all angles made on it is 180Â°
â‡’ âˆ AEP + âˆ AEQ = 180Â°
â‡’ 50Â° + x = 180Â°
â‡’ x = 180Â° – 50Â° = 130Â°
â‡’ x = 130Â° –eq(ii)
Now, from eq(i) and eq(ii) We conclude that x = y
As they are pair of alternate interior angles
So, AB || CD
Hence proved!!!
Question 2. In given figure, if AB || CD, CD || EF and y : z = 3 : 7, find x.
Solution:
Given AB || CD and CD || EF y : z = 3 : 7
â‡’ AB || CD || EF
â‡’ AB || EF
So, x=z [alternate interior angles] –eq(i)
Again AB || CD
â‡’ x + y = 180Â° [Co-interior angles]
â‡’ z + y = 180Â° –eq(ii) [from eq(i)]
But given that y : z = 3 : 7
â‡’ z = (7/3) y = (7/3)(180Â° – z) [from eq(ii)]
â‡’ 10z = 7 * 180Â°
â‡’ z = (7 * 180Â°)/10 =126Â°
â‡’ z = 126Â° –eq(iii)
from eq(i) and eq(iii) we have
â‡’ x = 126Â°
Question 3. In given figure, if AB || CD, EF âŠ¥ CD and âˆ GED = 126Â°, find âˆ AGE, âˆ GEF and âˆ FGE.
Solution:
Given AB || CD, EF âŠ¥ CD, âˆ GED = 126Â° and âˆ FED =90Â°
â‡’ âˆ GED = âˆ GEF + âˆ FED
â‡’ 126Â° = âˆ GEF + 90Â° [Given]
â‡’ âˆ GEF = 36Â°
As, AB || CD and GE is a transversal
So, âˆ FGE + âˆ GED = 180Â° [ sum of Co- interior angles is 180 ]
â‡’ âˆ FGE + 126Â° = 180Â° [ Given ]
â‡’ âˆ FGE = 54Â°
As, AB || CD and GE is a transversal
So, âˆ AGE = âˆ GED [ alternate angles are equal ]
â‡’ âˆ AGE = 126Â°
Question 4. In given figure, if PQ || ST, âˆ PQR = 110Â° and âˆ RST = 130Â°, find âˆ QRS.
[Hint: Draw a line parallel to ST through point R.]
Solution:
Firstly we have drawn a line EF parallel to ST (EF || ST)
Since, PQ || ST [Given] and EF || ST [ Construction ]
So, PQ || EF and QR is a transversal
â‡’ âˆ PQR = âˆ QRF [ Alternate interior angles ]
â‡’ âˆ QRF = 110Â° [Given âˆ PQR =110Â° ]
â‡’ âˆ QRF = âˆ QRS + âˆ SRF
â‡’ âˆ QRS + âˆ SRF = 110Â° –eq(i)
Again ST || EF and RS is a transversal
â‡’âˆ RST + âˆ SRF = 180Â° [ sum of Co-interior angles is 180Â° ]
â‡’130Â° + âˆ SRF =180Â° [Given]
â‡’âˆ SRF =50Â°
Now , from eq(i)
â‡’ âˆ QRS + âˆ SRF = 110Â°
â‡’ âˆ QRS +50 = 110Â°
â‡’ Thus, âˆ QRS = 60Â°
Question 5. In Fig. 6.32, if AB || CD, âˆ APQ = 50Â° and âˆ PRD = 127Â°, find x and y.
Solution:
Given AB || CD and PQ is a transversal
â‡’ âˆ APQ = âˆ PQR [ Alternate interior angles ]
â‡’ x= 50Â° [ Given âˆ APQ = 50Â° ]
â‡’ x = 50Â°
Again, AB || CD and PR is a transversal
â‡’âˆ APR = âˆ PRD [ Alternate interior angles ]
â‡’ âˆ APR = 127Â° [ Given âˆ PRD = 127Â° ]
â‡’ 50Â° + y =127Â° [Given âˆ APQ = 50Â° ]
â‡’ y =127Â° – 50Â° = 77Â°
â‡’ y = 77Â°
Thus, x = 50Â° and y = 77Â°
Question 6. In given figure, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.
Solution:
Draw ray BL âŠ¥ PQ and CM âŠ¥ RS
Since, PQ || RS â‡’ BL || CM
â‡’[ So, BL || PQ and CM || RS ]
Now, BL || CM and BC is a transversal
â‡’ âˆ LBC = âˆ MCB –eq(i) [ Alternate interior angles ]
Since, angle of incidence = angle of reflection
â‡’ âˆ ABL = âˆ LBC and âˆ MCB = âˆ MCD
â‡’ âˆ ABL = âˆ MCD –eq(ii) [By eq(i)]
â‡’Adding eq(i) and eq(ii) we get
â‡’ âˆ LBC + âˆ ABL = âˆ MCB + âˆ MCD
â‡’ âˆ ABC = âˆ BCD
i.e, a pair of alternate angles are equal
Thus, AB || CD
Hence, Proved !!!
Lines and Angles – Exercise 6.3
Question 1: In Figure, sides QP and RQ of Î”PQR are produced to points S and T respectively. If âˆ SPR = 135Â° and âˆ PQT = 110Â°, find âˆ PRQ.
Solution:
Given: âˆ TQP = 110Â°, âˆ SPR = 135Â°
TQR is a Straight line as we can see in the figure
As we have studied in this chapter, TQP and PQR will form a linear pair
â‡’ âˆ TQP + âˆ PQR = 180Â° ———-(i)
Putting the value of âˆ TQP = 110Â° in Equation (i) we get,
â‡’ 110Â° + âˆ PQR = 180Â°
â‡’ âˆ PQR = 70Â°
Consider the Î”PQR,
Here, the side QP is extended to S and so, SPR forms the exterior angle.
Thus, âˆ SPR (âˆ SPR = 135Â°) is equal to the sum of interior opposite angles. (Triangle property)
Or, âˆ PQR + âˆ PRQ = 135Â° ———(ii)
Now, putting the value of PQR = 70Â° in equation (ii) we get,
âˆ PRQ = 135Â° – 70Â°
Hence, âˆ PRQ = 65Â°
Question 2: In Figure, âˆ X = 62Â°, âˆ XYZ = 54Â°. If YO and ZO are the bisectors of XYZ and XZY respectively of Î” XYZ, find OZY and YOZ.
Solution:
Given: âˆ X = 62Â°, âˆ XYZ = 54Â°
As we have studied in this chapter,
We know that the sum of the interior angles of the triangle is 180Â°.
So, âˆ X +âˆ XYZ +âˆ XZY = 180Â°
Putting the values as given in the question we get,
62Â°+54Â° + âˆ XZY = 180Â°
Or, âˆ XZY = 64Â°
Now, we know that ZO is the bisector so,
âˆ OZY = Â½ XZY
âˆ´ âˆ OZY = 32Â°
Similarly, YO is a bisector and so,
âˆ OYZ = Â½ XYZ
Or, âˆ OYZ = 27Â° (As XYZ = 54Â°)
Now, as the sum of the interior angles of the triangle,
âˆ OZY +âˆ OYZ +O = 180Â°
Putting their respective values, we get,
âˆ O = 180Â°-32Â°-27Â°
Hence, âˆ O = 121Â°
Question 3: In Figure, if AB || DE, âˆ BAC = 35Â° and âˆ CDE = 53Â°, find âˆ DCE.
Solution:
Given: AB || DE, âˆ BAC = 35Â° and âˆ CDE = 53Â°
Since, we know that AE is a transversal of AB and DE
Here, BAC and AED are alternate interior angles.
Hence, âˆ BAC = âˆ AED
âˆ BAC = 35Â° (Given)
âˆ AED = 35Â°
Now consider the triangle CDE. We know that the sum of the interior angles of a triangle is 180Â°.
âˆ´ âˆ DCE + âˆ CED + âˆ CDE = 180Â°
Putting the values, we get
âˆ DCE + 35Â° + 53Â° = 180Â°
Hence, âˆ DCE = 92Â°
Question 4: In Figure, if lines PQ and RS intersect at point T, such that âˆ PRT = 40Â°, âˆ RPT = 95Â° and âˆ TSQ = 75Â°, find âˆ SQT.
Solution:
Given: âˆ PRT = 40Â°, âˆ RPT = 95Â° and âˆ TSQ = 75Â°
In â–³PRT.
âˆ PRT +âˆ RPT + âˆ PTR = 180Â° (The Sum of all the angles of Triangle is 180Â°)
â‡’ âˆ PTR = 45Â°
Now âˆ PTR will be equal to STQ as they are vertically opposite angles.
â‡’ âˆ PTR = âˆ STQ = 45Â°
Again, in triangle STQ,
â‡’ âˆ TSQ +âˆ PTR + âˆ SQT = 180Â° (The Sum of all the angles of Triangle is 180Â°)
Solving this we get,
â‡’ âˆ SQT = 60Â°
Question 5: In Figure, if PQ âŠ¥ PS, PQ || SR, âˆ SQR = 28Â° and âˆ QRT = 65Â°, then find the values of x and y.
Solution:
Given: PQ âŠ¥ PS, PQ || SR, âˆ SQR = 28Â° and âˆ QRT = 65Â°
x + SQR = QRT (As they are alternate angles since QR is transversal)
Now, Putting the value of âˆ SQR = 28Â° and âˆ QRT = 65Â°
â‡’ x + 28Â° = 65Â°
âˆ´ x = 37Â°
It is also known that alternate interior angles are same and
â‡’ QSR = x = 37Â°
Also,
â‡’ QRS + QRT = 180Â° (As they form a Linear pair)
Putting the value of âˆ QRT = 65Â° we get,
â‡’ QRS + 65Â° = 180Â°
â‡’ QRS = 115Â°
As we know that the sum of the angles in a quadrilateral is 360Â°.
â‡’ P + Q + R + S = 360Â°
Putting their respective values, we get,
â‡’ S = 360Â° – 90Â° – 65Â° – 115Â° = 900
In Î” SPQ
â‡’ âˆ SPQ + x + y = 1800
â‡’ 900 + 370 + y = 1800
â‡’ y = 1800 â€“ 1270 = 530
Hence, y = 53Â°
Question 6: In Figure, the side QR of Î”PQR is produced to a point S. If the bisectors of âˆ PQR and âˆ PRS meet at point T, then prove that âˆ QTR = Â½ âˆ QPR.
Solution:
Given: T is the bisector of âˆ PQR and âˆ PRS,
To Prove: âˆ QTR = Â½ âˆ QPR
Proof:
Consider the Î”PQR.
âˆ PRS is an exterior angle.
âˆ QPR and âˆ PQR are interior angles.
â‡’ âˆ PRS = âˆ QPR + âˆ PQR (According to triangle property)
â‡’ âˆ PRS – âˆ PQR = âˆ QPR ————(i)
Now, consider the Î”QRT,
âˆ TRS = âˆ TQR + âˆ QTR (Since exterior angle are equal)
â‡’ âˆ QTR = âˆ TRS – âˆ TQR
We know that QT and RT bisect âˆ PQR and âˆ PRS respectively.
So, âˆ PRS = 2 âˆ TRS and âˆ PQR = 2âˆ TQR
Now,
â‡’ âˆ QTR = Â½ âˆ PRS â€“ Â½ âˆ PQR
â‡’ âˆ QTR = Â½ âˆ (PRS – PQR)
From equation (i) we know that âˆ PRS – âˆ PQR = âˆ QPR,
â‡’ âˆ QTR = Â½ âˆ QPR
Hence, Proved.
Important Points to Remember:
- NCERT Solutions provides accurate and complete solutions for problems given in NCERT textbooks.
- These solutions are given in step by step manner for a better understanding of the problems.
- NCERT Solutions for Class 9 will help students to learn the solution for all the NCERT Problems.
FAQs on NCERT Solutions for Class 9 Maths Chapter 6 â€“ Lines and Angles
1. Why is it important to learn about the lines and angles?
Lines and Angles are fundamental concepts in geometry. They form the building blocks for understanding the properties and relationships of shapes and figures. By studying lines and angles, students develop a strong geometric foundation and gain insights into the properties of various geometric shapes, such as triangles, quadrilaterals, and circles.
2. What topics are covered in NCERT Solutions for Chapter 6 â€“ Lines and Angles?
NCERT Solutions for Class 9 Maths Chapter 6 â€“ Lines and Angles covers topics such as characteristics of the angles, kinds of angles, pairs of angles, transversals, parallel lines, the angle sum property of triangles, and axioms.
3. How can NCERT Solutions for Class 9 Maths Chapter 6 â€“ Lines and Angles help me?
NCERT Solutions for Class 9 Maths Chapter 6 â€“ Lines and Angles can help you solve the NCERT exercise without any limitations. If you are stuck on a problem you can find its solution in these solutions and free yourself from the frustration of being stuck on some question.
4. How many exercises are there in Class 9 Maths Chapter 6 â€“ Lines and Angles?
There are 3 exercises in the Class 9 Maths Chapter 6 â€“ Lines and Angles which covers all the important topics and sub-topics.
5. Where can I find NCERT Solutions for Class 9 Maths Chapter 6 â€“ Lines and Angles?
You can find these NCERT Solutions in this article created by our team of experts at GeeksforGeeks.