NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles Ex 6.3 are part of NCERT Solutions for Class 9 Maths. Here we have given NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles Ex 6.3.
| Board | CBSE |
| Textbook | NCERT |
| Class | Class 9 |
| Subject | Maths |
| Chapter | Chapter 6 |
| Chapter Name | Lines and Angles |
| Exercise | Ex 6.3 |
| Number of Questions Solved | 6 |
| Category | NCERT Solutions |
NCERT Solutions for Class 9 Maths Chapter 6 Lines and Angles Ex 6.3
Ex 6.3 Class 9 Maths Question 1.
In the given figure, sides QP and RQ of ΔPQR are produced to points S and T respectively.
If ∠SPR = 135° and ∠PQT = 110°, find ∠PRQ.
Solution:
∠RPS + ∠RPQ = 180 °
135° + ∠RPQ = 180° [∵ ∠RPS = 135° (given)]
⇒ RPQ = 180°-135° = 45°
Now, ∠RPQ + ∠PRQ = ∠PQT [ext. angle = sum of int. opp. angles]
⇒ 45° + ∠PRQ = 110°
⇒ ∠PRQ = 110° – 45° = 65°
Ex 6.3 Class 9 Maths Question 2.
In the given figure, ∠X= 62°,∠XYZ= 54°. If YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively of ΔXYZ, find ∠OZY and ∠YOZ.
Solution:
Given, ∠X = 62° and ∠XYZ = 54°
In ΔXYZ, ∠XYZ + ∠YXZ + ∠XZY = 180° [since a sum of all the angles of a triangle is 180°]
54°+62°+∠180°⇒ 116°+∠XYZ =18O°
∠XZY=180°-116°
∴ ∠XZY = 64°
Also, given that YO and ZO are the bisectors of ∠XYZ and ∠XZY, respectively.
So, ∠OYZ = \(\cfrac { 1 }{ 2 }\) ∠XYZ => ∠OYZ = \(\cfrac { 1 }{ 2 }\) x 54° = 27°
and ∠OZY = \(\cfrac { 1 }{ 2 }\) ∠XZY= \(\cfrac { 1 }{ 2 }\) x 64°= ∠OZY=32°
Now, in ΔOYZ, ∠OYZ + ∠OZY + ∠YOZ = 180° [since, sum of all the angles of a triangle Is 180°]
27° + 32° + ∠YOZ = 180°
59°+∠YOZ = 180° ⇒ ∠YOZ = 180°—59°
∴ ∠YOZ = 121°
Hence,∠OZY = 32° and ∠YOZ = 121°
Ex 6.3 Class 9 Maths Question 3.
In figure,If AB DE, ∠BAC= 35°and ∠CDE= 53°,find ∠DCE.
Solution:
We have, AB || DE
⇒ ∠AED = ∠ BAE (Alternate interior angles)
Now, ∠BAE = ∠BAC
⇒ ∠BAE=35° [ ∵ ∠BAC = 35°Given)]
∠AED = 35°
In Δ DCE,
∴ ∠DCE + ∠CED+ EDC = 180°, (∵ Sum of all angles of triangle is equal to 180°)
⇒ ∠DCE + 35° + 53° – 180° (∵∠AED = ∠CED = 35°)
⇒ ∠DCE = 180 °-(35° + 53° ) ⇒ ∠DCE = 92°
Ex 6.3 Class 9 Maths Question 4.
In the given figure, if lines PQ and RS intersect at point T such that ∠PRT = 40°, ∠RPT = 95°
and ∠TSQ = 75° then find ∠SQT.
Solution:
Given, ∠PRT = 40°, ∠RPT = 95° and
∠TSQ = 75°
In Δ PRT, ∠PRT + ∠RPT + ∠PTR = 180° …(i) [since, sum of all the angles of a triangle is 180°]
On putting ∠PRT =40° and ∠RPT = 95° in eq. (i), we get
40° + 95° + ∠PTR = 180°
⇒ 135° + ∠PTR = 180° ⇒ ∠PTR = 180° -135° ⇒ ∠PTR = 45°
Now, ∠PTR = ∠QTS [vertically opposite angles]
∴ ∠QTS – 45°
In ∠TQS, ∠QTS + ∠TSQ + ∠TQS = 180° … (ii)
[since, sum of all the angles of a triangle is 180°]
On putting ∠QTS = 45° and ∠TSQ = 75° in eq. (ii), we get
45° + 75° + ∠TQS = 180° ⇒120° + ∠TQS = 180°
⇒ ∠TQS – 180°-120°
∴ ∠TQS =60°
Hence,∠TQS = ∠SQT = 60°
Ex 6.3 Class 9 Maths Question 5.
In the given figure, if PQ ⊥ PS, PQ || SR, ∠SQR – 28° and ∠QRT – 65° then find the values of x and y.
Solution:
For Δ QSR, ∠QRT is an exterior angle
∠QRT = ∠SQR + ∠QSR [∵ exterior angle = sum of interior opposite angles]
⇒ 65° = 28° + ∠QSR [∵ ∠QRT = 65° and ∠SQR = 28°, given]
∠QSR = 65° – 28° ⇒ ∠QSR = 37°
Given PQ || SR and SQ is the transversal which intersects PQ and ST at Q and S, respectively.
∴ ∠QSR = ∠PQS [alternate interior angles]
⇒ x = 37°
Now, in ΔPQS, ∠SPQ + ∠PQS + ∠PSQ = 180°[since, sum of all the angles of a triangle is 180°]
⇒ 90° + 37° + y = 180° [∵ PQ ⊥ PS ⇒ ∠SPQ = 90°]
⇒ 127° + y= 180°⇒ y = 180° -127° = 53°
Hence, x = 37° and y= 53°.
Ex 6.3 Class 9 Maths Question 6.
In the given figure, the side QR of ΔPQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR = \(\cfrac { 1 }{ 2 }\) ∠QPR.
Solution:
In ΔTQR, ∠QTR + ∠TQR + ∠QRT = 1800 [sum of all the angles of a triangle is 180°]
∠QRT = 1800 – [∠TQR + ∠QRT]
∠PQR + ∠QRP + ∠PRT
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